A127093 -id:A127093 - cp's OEIS frontend

A000203 a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144

Offset: 1

N. J. A. Sloane

Multiplicative: If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (this sequence) (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
A number n is abundant if sigma(n) > 2n (cf. A005101), perfect if sigma(n) = 2n (cf. A000396), deficient if sigma(n) < 2n (cf. A005100).
a(n) is the number of sublattices of index n in a generic 2-dimensional lattice. - Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001 [In the language of group theory, a(n) is the number of index-n subgroups of Z x Z. - Jianing Song, Nov 05 2022]
The sublattices of index n are in one-to-one correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} d = sigma(n), which is a(n). A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * Product_{p|n} (1+1/p), which is A001615. [Cf. Grady reference.]
Sum of number of common divisors of n and m, where m runs from 1 to n. - Naohiro Nomoto, Jan 10 2004
a(n) is the cardinality of all extensions over Q_p with degree n in the algebraic closure of Q_p, where p>n. - Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004. Cf. A100976, A100977, A100978 (p-adic extensions).
Let s(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) + a(n-15) - a(n-22) - a(n-26) + ..., then a(n) = s(n) if n is not pentagonal, i.e., n != (3 j^2 +- j)/2 (cf. A001318), and a(n) is instead s(n) - ((-1)^j)*n if n is pentagonal. - Gary W. Adamson, Oct 05 2008 [corrected Apr 27 2012 by William J. Keith based on Ewell and by Andrey Zabolotskiy, Apr 08 2022]
Write n as 2^k * d, where d is odd. Then a(n) is odd if and only if d is a square. - Jon Perry, Nov 08 2012
Also total number of parts in the partitions of n into equal parts. - Omar E. Pol, Jan 16 2013
Note that sigma(3^4) = 11^2. On the other hand, Kanold (1947) shows that the equation sigma(q^(p-1)) = b^p has no solutions b > 2, q prime, p odd prime. - N. J. A. Sloane, Dec 21 2013, based on postings to the Number Theory Mailing List by Vladimir Letsko and Luis H. Gallardo
Limit_{m->infinity} (Sum_{n=1..prime(m)} a(n)) / prime(m)^2 = zeta(2)/2 = Pi^2/12 (A072691). See more at A244583. - Richard R. Forberg, Jan 04 2015
a(n) + A000005(n) is an odd number iff n = 2m^2, m>=1. - Richard R. Forberg, Jan 15 2015
a(n) = a(n+1) for n = 14, 206, 957, 1334, 1364 (A002961). - Zak Seidov, May 03 2016
Equivalent to the Riemann hypothesis: a(n) < H(n) + exp(H(n))*log(H(n)), for all n>1, where H(n) is the n-th harmonic number (Jeffrey Lagarias). See A057641 for more details. - Ilya Gutkovskiy, Jul 05 2016
a(n) is the total number of even parts in the partitions of 2*n into equal parts. More generally, a(n) is the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts (the comment dated Jan 16 2013 is the case for k = 1). - Omar E. Pol, Nov 18 2019
From Jianing Song, Nov 05 2022: (Start)
a(n) is also the number of order-n subgroups of C_n X C_n, where C_n is the cyclic group of order n. Proof: by the correspondence theorem in the group theory, there is a one-to-one correspondence between the order-n subgroups of C_n X C_n = (Z x Z)/(nZ x nZ) and the index-n subgroups of Z x Z containing nZ x nZ. But an index-n normal subgroup of a (multiplicative) group G contains {g^n : n in G} automatically. The desired result follows from the comment from Naohiro Nomoto above.
The number of subgroups of C_n X C_n that are isomorphic to C_n is A001615(n). (End)

			For example, 6 is divisible by 1, 2, 3 and 6, so sigma(6) = 1 + 2 + 3 + 6 = 12.
Let L = <V,W> be a 2-dimensional lattice. The 7 sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V,2W>, <2V+W,2W>, <2V,2W+V>. Compare A001615.

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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 2nd formula.
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Ross Honsberger, "Mathematical Gems, Number One," The Dolciani Mathematical Expositions, Published and Distributed by The Mathematical Association of America, page 116.
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G. Pólya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ Press 1954, page 92.
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Robert M. Young, Excursions in Calculus, The Mathematical Association of America, 1992 p. 361.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 20000 terms from N. J. A. Sloane)
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B. Apostol and L. Petrescu, Extremal Orders of Certain Functions Associated with Regular Integers (mod n), Journal of Integer Sequences, 2013, # 13.7.5.
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
M. Baake and U. Grimm, Quasicrystalline combinatorics
Henry Bottomley, Illustration of initial terms
C. K. Caldwell, The Prime Glossary, sigma function
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
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J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012. - _N. J. A. Sloane_, Dec 25 2012
R. L. Duncan, Note on the Divisors of a Number, The American Mathematical Monthly, 68(4), 356-359, (1961).
Jason Earls, The Smarandache sum of composites between factors function, in Smarandache Notions Journal (2004), Vol. 14.1, page 243.
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L. Euler, Observatio de summis divisorum
L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009.
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Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
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Johan Gielis and Ilia Tavkhelidze, The general case of cutting of GML surfaces and bodies, arXiv:1904.01414 [math.GM], 2019.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
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Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], 2022.
Douglas E. Iannucci, On sums of the small divisors of a natural number, arXiv:1910.11835 [math.NT], 2019.
Antti Karttunen, Scheme program for computing this sequence.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math.CO/0503436, 2005.
K. Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)
Walter Nissen, Abundancy : Some Resources
P. Pollack and C. Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26; errata.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp. 83 (2014), 1903-1913.
John S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. (1992) A48, 500-508. - _N. J. A. Sloane_, Mar 14 2009
John S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices II, Acta Cryst. A49 (1993), 293-300. - _N. J. A. Sloane_, Mar 14 2009
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - _N. J. A. Sloane_, Feb 23 2009
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Eric Weisstein's World of Mathematics, Divisor Function
Wikipedia, Divisor function
Index entries for sequences related to sublattices
Index entries for sequences related to sigma(n)
Index entries for "core" sequences

See A034885, A002093 for records. Bisections give A008438, A062731. Values taken are listed in A007609. A054973 is an inverse function.
For partial sums see A024916.
Row sums of A127093.
sigma_i (i=0..25): A000005, A000203, A001157, A001158, A001159, A001160, A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972, A281959.
Cf. A144736, A158951, A158902, A174740, A147843, A001158, A001160, A001065, A002192, A001001, A001615 (primitive sublattices), A039653, A088580, A074400, A083728, A006352, A002659, A083238, A000593, A050449, A050452, A051731, A027748, A124010, A069192, A057641, A001318.
Cf. A009194, A082062 (gcd(a(n),n) and its largest prime factor), A179931, A192795 (gcd(a(n),A001157(n)) and largest prime factor).
Cf. also A034448 (sum of unitary divisors).
Cf. A007955 (products of divisors).
Cf. A144613, A144614, A144615, A146076.
A001227, A000593 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
Cf. A001221, A054533.

A000203:=List([1..10^2],n->Sigma(n)); # Muniru A Asiru, Oct 01 2017

a000203 n = product $ zipWith (\p e -> (p^(e+1)-1) `div` (p-1)) (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, May 07 2012

Magma
```
[SumOfDivisors(n): n in [1..70]];
```

[DivisorSigma(1,n): n in [1..70]]; // Bruno Berselli, Sep 09 2015

with(numtheory): A000203 := n->sigma(n); seq(A000203(n), n=1..100);

Table[ DivisorSigma[1, n], {n, 100}]
a[ n_] := SeriesCoefficient[ QPolyGamma[ 1, 1, q] / Log[q]^2, {q, 0, n}]; (* Michael Somos, Apr 25 2013 *)

makelist(divsum(n),n,1,1000); /* Emanuele Munarini, Mar 26 2011 */

numlib::sigma(n)$ n=1..81 // Zerinvary Lajos, May 13 2008

PARI
```
{a(n) = if( n<1, 0, sigma(n))};
```

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) /(1 - p*X))[n])};

{a(n) = if( n<1, 0, polcoeff( sum( k=1, n, x^k / (1 - x^k)^2, x * O(x^n)), n))}; /* Michael Somos, Jan 29 2005 */

max_n = 30; ser = - sum(k=1,max_n,log(1-x^k)); a(n) = polcoeff(ser,n)*n \\ Gottfried Helms, Aug 10 2009

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 1)
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Jan 03 2021

from math import prod
from sympy import factorint
def a(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Feb 25 2024
(APL, Dyalog dialect) A000203 ← +/{ð←⍵{(0=⍵|⍺)/⍵}⍳⌊⍵*÷2 ⋄ 1=⍵:ð ⋄ ð,(⍵∘÷)¨(⍵=(⌊⍵*÷2)*2)↓⌽ð} ⍝ Antti Karttunen, Feb 20 2024

[sigma(n, 1) for n in range(1, 71)]  # Zerinvary Lajos, Jun 04 2009

(definec (A000203 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (/ (- (expt p (+ 1 e)) 1) (- p 1)) (A000203 (A028234 n)))))) ;; Uses macro definec from http://oeis.org/wiki/Memoization#Scheme - Antti Karttunen, Nov 25 2017

(define (A000203 n) (let ((r (sqrt n))) (let loop ((i (inexact->exact (floor r))) (s (if (integer? r) (- r) 0))) (cond ((zero? i) s) ((zero? (modulo n i)) (loop (- i 1) (+ s i (/ n i)))) (else (loop (- i 1) s)))))) ;; (Stand-alone program) - Antti Karttunen, Feb 20 2024

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1). - David W. Wilson, Aug 01 2001
For the following bounds and many others, see Mitrinovic et al. - N. J. A. Sloane, Oct 02 2017
If n is composite, a(n) > n + sqrt(n).
a(n) < n*sqrt(n) for all n.
a(n) < (6/Pi^2)*n^(3/2) for n > 12.
G.f.: -x*deriv(eta(x))/eta(x) where eta(x) = Product_{n>=1} (1-x^n). - Joerg Arndt, Mar 14 2010
L.g.f.: -log(Product_{j>=1} (1-x^j)) = Sum_{n>=1} a(n)/n*x^n. - Joerg Arndt, Feb 04 2011
Dirichlet convolution of phi(n) and tau(n), i.e., a(n) = sum_{d|n} phi(n/d)*tau(d), cf. A000010, A000005.
a(n) is odd iff n is a square or twice a square. - Robert G. Wilson v, Oct 03 2001
a(n) = a(n*prime(n)) - prime(n)*a(n). - Labos Elemer, Aug 14 2003 (Clarified by Omar E. Pol, Apr 27 2016)
a(n) = n*A000041(n) - Sum_{i=1..n-1} a(i)*A000041(n-i). - Jon Perry, Sep 11 2003
a(n) = -A010815(n)*n - Sum_{k=1..n-1} A010815(k)*a(n-k). - Reinhard Zumkeller, Nov 30 2003
a(n) = f(n, 1, 1, 1), where f(n, i, x, s) = if n = 1 then s*x else if p(i)|n then f(n/p(i), i, 1+p(i)*x, s) else f(n, i+1, 1, s*x) with p(i) = i-th prime (A000040). - Reinhard Zumkeller, Nov 17 2004
Recurrence: n^2*(n-1)*a(n) = 12*Sum_{k=1..n-1} (5*k*(n-k) - n^2)*a(k)*a(n-k), if n>1. - Dominique Giard (dominique.giard(AT)gmail.com), Jan 11 2005
G.f.: Sum_{k>0} k * x^k / (1 - x^k) = Sum_{k>0} x^k / (1 - x^k)^2. Dirichlet g.f.: zeta(s)*zeta(s-1). - Michael Somos, Apr 05 2003. See the Hardy-Wright reference, p. 312. first equation, and p. 250, Theorem 290. - Wolfdieter Lang, Dec 09 2016
For odd n, a(n) = A000593(n). For even n, a(n) = A000593(n) + A074400(n/2). - Jonathan Vos Post, Mar 26 2006
Equals the inverse Moebius transform of the natural numbers. Equals row sums of A127093. - Gary W. Adamson, May 20 2007
A127093 * [1/1, 1/2, 1/3, ...] = [1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, ...]. Row sums of triangle A135539. - Gary W. Adamson, Oct 31 2007
a(n) = A054785(2*n) - A000593(2*n). - Reinhard Zumkeller, Apr 23 2008
a(n) = n*Sum_{k=1..n} A060642(n,k)/k*(-1)^(k+1). - Vladimir Kruchinin, Aug 10 2010
Dirichlet convolution of A037213 and A034448. - R. J. Mathar, Apr 13 2011
G.f.: A(x) = x/(1-x)*(1 - 2*x*(1-x)/(G(0) - 2*x^2 + 2*x)); G(k) = -2*x - 1 - (1+x)*k + (2*k+3)*(x^(k+2)) - x*(k+1)*(k+3)*((-1 + (x^(k+2)))^2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2011
a(n) = A001065(n) + n. - Mats Granvik, May 20 2012
a(n) = A006128(n) - A220477(n). - Omar E. Pol, Jan 17 2013
a(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*A196020(n,k). - conjectured by Omar E. Pol, Feb 02 2013, and proved by Max Alekseyev, Nov 17 2013
a(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*A000330(k)*A000716(n-A000217(k)). - Mircea Merca, Mar 05 2014
a(n) = A240698(n, A000005(n)). - Reinhard Zumkeller, Apr 10 2014
a(n) = Sum_{d^2|n} A001615(n/d^2) = Sum_{d^3|n} A254981(n/d^3). - Álvar Ibeas, Mar 06 2015
a(3*n) = A144613(n). a(3*n + 1) = A144614(n). a(3*n + 2) = A144615(n). - Michael Somos, Jul 19 2015
a(n) = Sum{i=1..n} Sum{j=1..i} cos((2*Pi*n*j)/i). - Michel Lagneau, Oct 14 2015
a(n) = A000593(n) + A146076(n). - Omar E. Pol, Apr 05 2016
a(n) = A065475(n) + A048050(n). - Omar E. Pol, Nov 28 2016
a(n) = (Pi^2*n/6)*Sum_{q>=1} c_q(n)/q^2, with the Ramanujan sums c_q(n) given in A054533 as a c_n(k) table. See the Hardy reference, p. 141, or Hardy-Wright, Theorem 293, p. 251. - Wolfdieter Lang, Jan 06 2017
G.f. also (1 - E_2(q))/24, with the g.f. E_2 of A006352. See e.g., Hardy, p. 166, eq. (10.5.5). - Wolfdieter Lang, Jan 31 2017
From Antti Karttunen, Nov 25 2017: (Start)
a(n) = A048250(n) + A162296(n).
a(n) = A092261(n) * A295294(n). [This can be further expanded, see comment in A291750.] (End)
a(n) = A000593(n) * A038712(n). - Ivan N. Ianakiev and Omar E. Pol, Nov 26 2017
a(n) = Sum_{q=1..n} c_q(n) * floor(n/q), where c_q(n) is the Ramanujan's sum function given in A054533. - Daniel Suteu, Jun 14 2018
a(n) = Sum_{k=1..n} gcd(n, k) / phi(n / gcd(n, k)), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 21 2018
a(n) = (2^(1 + (A000005(n) - A001227(n))/(A000005(n) - A183063(n))) - 1)*A000593(n) = (2^(1 + (A183063(n)/A001227(n))) - 1)*A000593(n). - Omar E. Pol, Nov 03 2018
a(n) = Sum_{i=1..n} tau(gcd(n, i)). - Ridouane Oudra, Oct 15 2019
From Peter Bala, Jan 19 2021: (Start)
G.f.: A(x) = Sum_{n >= 1} x^(n^2)*(x^n + n*(1 - x^(2*n)))/(1 - x^n)^2 - differentiate equation 5 in Arndt w.r.t. x, and set x = 1.
A(x) = F(x) + G(x), where F(x) is the g.f. of A079667 and G(x) is the g.f. of A117004. (End)
a(n) = Sum_{k=1..n} tau(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
With the convention that a(n) = 0 for n <= 0 we have the recurrence a(n) = t(n) + Sum_{k >= 1} (-1)^(k+1)*(2*k + 1)*a(n - k*(k + 1)/2), where t(n) = (-1)^(m+1)*(2*m+1)*n/3 if n = m*(m + 1)/2, with m positive, is a triangular number else t(n) = 0. For example, n = 10 = (4*5)/2 is a triangular number, t(10) = -30, and so a(10) = -30 + 3*a(9) - 5*a(7) + 7*a(4) = -30 + 39 - 40 + 49 = 18. - Peter Bala, Apr 06 2022
Recurrence: a(p^x) = p*a(p^(x-1)) + 1, if p is prime and for any integer x. E.g., a(5^3) = 5*a(5^2) + 1 = 5*31 + 1 = 156. - Jules Beauchamp, Nov 11 2022
Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/24 - 1/(8*Pi) = A319462. - Vaclav Kotesovec, May 07 2023
a(n) < (7n*A001221(n) + 10*n)/6 [Duncan, 1961] (see Duncan and Tattersall). - Stefano Spezia, Jul 13 2025

A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9, 2, 8, 2, 8

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is Product p^e(p) then d(n) = Product (e(p) + 1). More generally, for k > 0, sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1) is the sum of the k-th powers of the divisors of n.
Number of ways to write n as n = x*y, 1 <= x <= n, 1 <= y <= n. For number of unordered solutions to x*y=n, see A038548.
Note that d(n) is not the number of Pythagorean triangles with radius of the inscribed circle equal to n (that is A078644). For number of primitive Pythagorean triangles having inradius n, see A068068(n).
Number of factors in the factorization of the polynomial x^n-1 over the integers. - T. D. Noe, Apr 16 2003
Also equal to the number of partitions p of n such that all the parts have the same cardinality, i.e., max(p)=min(p). - Giovanni Resta, Feb 06 2006
Equals A127093 as an infinite lower triangular matrix * the harmonic series, [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, May 10 2007
For odd n, this is the number of partitions of n into consecutive integers. Proof: For n = 1, clearly true. For n = 2k + 1, k >= 1, map each (necessarily odd) divisor to such a partition as follows: For 1 and n, map k + (k+1) and n, respectively. For any remaining divisor d <= sqrt(n), map (n/d - (d-1)/2) + ... + (n/d - 1) + (n/d) + (n/d + 1) + ... + (n/d + (d-1)/2) {i.e., n/d plus (d-1)/2 pairs each summing to 2n/d}. For any remaining divisor d > sqrt(n), map ((d-1)/2 - (n/d - 1)) + ... + ((d-1)/2 - 1) + (d-1)/2 + (d+1)/2 + ((d+1)/2 + 1) + ... + ((d+1)/2 + (n/d - 1)) {i.e., n/d pairs each summing to d}. As all such partitions must be of one of the above forms, the 1-to-1 correspondence and proof is complete. - Rick L. Shepherd, Apr 20 2008
Number of subgroups of the cyclic group of order n. - Benoit Jubin, Apr 29 2008
Equals row sums of triangle A143319. - Gary W. Adamson, Aug 07 2008
Equals row sums of triangle A159934, equivalent to generating a(n) by convolving A000005 prefaced with a 1; (1, 1, 2, 2, 3, 2, ...) with the INVERTi transform of A000005, (A159933): (1, 1,-1, 0, -1, 2, ...). Example: a(6) = 4 = (1, 1, 2, 2, 3, 2) dot (2, -1, 0, -1, 1, 1) = (2, -1, 0, -2, 3, 2) = 4. - Gary W. Adamson, Apr 26 2009
Number of times n appears in an n X n multiplication table. - Dominick Cancilla, Aug 02 2010
Number of k >= 0 such that (k^2 + k*n + k)/(k + 1) is an integer. - Juri-Stepan Gerasimov, Oct 25 2015
The only numbers k such that tau(k) >= k/2 are 1,2,3,4,6,8,12. - Michael De Vlieger, Dec 14 2016
a(n) is also the number of partitions of 2*n into equal parts, minus the number of partitions of 2*n into consecutive parts. - Omar E. Pol, May 03 2017
From Tomohiro Yamada, Oct 27 2020: (Start)
Let k(n) = log d(n)*log log n/(log 2 * log n), then lim sup k(n) = 1 (Hardy and Wright, Chapter 18, Theorem 317) and k(n) <= k(6983776800) = 1.537939... (the constant A280235) for every n (Nicolas and Robin, 1983).
There exist infinitely many n such that d(n) = d(n+1) (Heath-Brown, 1984). The number of such integers n <= x is at least c*x/(log log x)^3 (Hildebrand, 1987) but at most O(x/sqrt(log log x)) (Erdős, Carl Pomerance and Sárközy, 1987). (End)
Number of 2D grids of n congruent rectangles with two different side lengths, in a rectangle, modulo rotation (cf. A038548 for squares instead of rectangles). Also number of ways to arrange n identical objects in a rectangle (NOT modulo rotation, cf. A038548 for modulo rotation); cf. A007425 and A140773 for the 3D case. - Manfred Boergens, Jun 08 2021
The constant quoted above from Nicolas and Robin, 6983776800 = 2^5 * 3^3 * 5^2 * 7 * 11 * 13 * 17 * 19, appears arbitrary, but interestingly equals 2 * A095849(36). That second factor is highly composite and deeply composite. - Hal M. Switkay, Aug 08 2025

			G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 2*x^7 + 4*x^8 + 3*x^9 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 345, Exercise XXI(16). MR0121327 (22 #12066)
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 55.
G. H. Hardy and E. M. Wright, revised by D. R. Heath-Brown and J. H. Silverman, An Introduction to the Theory of Numbers, 6th ed., Oxford Univ. Press, 2008.
K. Knopp, Theory and Application of Infinite Series, Blackie, London, 1951, p. 451.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Chap. II. (For inequalities, etc.)
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. Has many references to this sequence. - N. J. A. Sloane, Jun 02 2014
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. Spearman and K. S. Williams, Handbook of Estimates in the Theory of Numbers, Carleton Math. Lecture Note Series No. 14, 1975; see p. 2.1.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 285.
E. C. Titchmarsh, The Theory of Functions, Oxford, 1938, p. 160.
Terence Tao, Poincaré's Legacies, Part I, Amer. Math. Soc., 2009, see pp. 31ff for upper bounds on d(n).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy, requires Flash plugin].
G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.1).
J. Bell, Lambert series in analytic number theory
R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340. [From _N. J. A. Sloane_, Mar 12 2009]
Henry Bottomley, Illustration of initial terms
D. M. Bressoud and M. V. Subbarao, On Uchimura's connection between partitions and the number of divisors, Can. Math. Bull. 27, 143-145 (1984). Zbl 0536.10013.
C. K. Caldwell, The Prime Glossary, Number of divisors
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
Jimmy Devillet and Gergely Kiss, Characterizations of biselective operations, arXiv:1806.02073 [math.RA], 2018.
P. Erdős and L. Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc. 2 (1952), pp. 257-271.
Paul Erdős, Carl Pomerance and András Sárközy, On locally repeated values of certain arithmetic functions, III, Proc. Amer. Math. Soc. 101 (1987), 1-7.
C. R. Fletcher, Rings of small order, Math. Gaz. vol. 64, p. 13, 1980.
Robbert Fokkink and Jan van Neerven, Problemen/UWC (in Dutch)
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.
D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31 (1984), 141-149.
Adolf Hildebrand, The divisor function at consecutive integers, Pacific J. Math. 129 (1987), 307-319.
J. J. Holt and J. W. Jones, Counting Divisors, Discovering Number Theory, Section 1.4.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.
R. G. Martinez, Jr., The Factor Zone, Number of Factors for 1 through 600.
Math Forum, Divisor Counting.
Mathematics Stack Exchange, A question on discrete Fourier Transform of some function
K. Matthews, Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n).
Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, corollary 2.1.
J. L. Nicolas and G. Robin, Majorations Explicites Pour le Nombre de Diviseurs de N, Canadian Mathematical Bulletin, Volume 26, Issue 4, 01 December 1983, pp. 485-492.
Matthew Parker, The first 25 million terms (7-Zip compressed file).
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003. [Cached copy, with permission (pdf only)]
Omar E. Pol, Illustration of initial terms: figure 1, figure 2, figure 3, figure 4, figure 5, (2009), figure 6 (a, b, c), (2013)
S. Ramanujan, On The Number Of Divisors Of A Number.
H. B. Reiter, Counting Divisors.
W. Sierpiński, Number Of Divisors And Their Sum.
Terence Tao, Poincaré's Legacies: pages from year two of a mathematical blog, see page 59.
E. C. Titchmarsh, On a series of Lambert type, J. London Math. Soc., 13 (1938), 248-253.
Keisuke Uchimura, An identity for the divisor generating function arising from sorting theory, J. Combin. Theory Ser. A 31 (1981), no. 2, 131--135. MR0629588 (82k:05015)
Wang Zheng Bing, Robert Fokkink and Wan Fokkink, A Relation Between Partitions and the Number of Divisors, Am. Math. Monthly, 102 (Apr., 1995), no. 4, 345-347.
Eric Weisstein's World of Mathematics, Binomial Number, Dirichlet Series Generating Function, Divisor Function, and Moebius Transform.
Wikipedia, Table of divisors.
Wolfram Research, Divisors of first 50 numbers
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

See A002183, A002182 for records. See A000203 for the sum-of-divisors function sigma(n).
For partial sums see A006218.
Cf. A007427 (Dirichlet Inverse), A001227, A001826, A001842, A005237, A005238, A006558, A006601, A019273, A027750, A034296, A039665, A049051, A049820, A051731, A061017, A066446, A091202, A091220, A106737, A115361, A127093, A129372, A129510, A143319, A156552, A159933, A159934, A163280, A183063, A237665, A263730.
Factorizations into given number of factors: writing n = x*y (A038548, unordered, A000005, ordered), n = x*y*z (A034836, unordered, A007425, ordered), n = w*x*y*z (A007426, ordered).
Cf. A000010, A095849.
Cf. A098198 (Dgf at s=2), A183030 (Dgf at s=3), A183031 (Dgf at s=3).

List([1..150],n->Tau(n)); # Muniru A Asiru, Mar 05 2019

divisors 1 = [1]
divisors n = (1:filter ((==0) . rem n)
               [2..n `div` 2]) ++ [n]
a = length . divisors
-- James Spahlinger, Oct 07 2012

a000005 = product . map (+ 1) . a124010_row  -- Reinhard Zumkeller, Jul 12 2013

function tau(n)
    i = 2; num = 1
    while i * i <= n
        if rem(n, i) == 0
            e = 0
            while rem(n, i) == 0
                e += 1
                n = div(n, i)
            end
            num *= e + 1
        end
        i += 1
    end
    return n > 1 ? num + num : num
end
println([tau(n) for n in 1:104])  # Peter Luschny, Sep 03 2023

[ NumberOfDivisors(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

with(numtheory): A000005 := tau; [ seq(tau(n), n=1..100) ];

Table[DivisorSigma[0, n], {n, 100}] (* Enrique Pérez Herrero, Aug 27 2009 *)
CoefficientList[Series[(Log[1 - q] + QPolyGamma[1, q])/(q Log[q]), {q, 0, 100}], q] (* Vladimir Reshetnikov, Apr 23 2013 *)
a[ n_] := SeriesCoefficient[ (QPolyGamma[ 1, q] + Log[1 - q]) / Log[q], {q, 0, Abs@n}]; (* Michael Somos, Apr 25 2013 *)
a[ n_] := SeriesCoefficient[ q/(1 - q)^2 QHypergeometricPFQ[ {q, q}, {q^2, q^2}, q, q^2], {q, 0, Abs@n}]; (* Michael Somos, Mar 05 2014 *)
a[n_] := SeriesCoefficient[q/(1 - q) QHypergeometricPFQ[{q, q}, {q^2}, q, q], {q, 0, Abs@n}] (* Mats Granvik, Apr 15 2015 *)
With[{M=500},CoefficientList[Series[(2x)/(1-x)-Sum[x^k (1-2x^k)/(1-x^k),{k,M}],{x,0,M}],x]] (* Mamuka Jibladze, Aug 31 2018 *)

numlib::tau (n)$ n=1..90 // Zerinvary Lajos, May 13 2008

{a(n) = if( n==0, 0, numdiv(n))}; /* Michael Somos, Apr 27 2003 */

{a(n) = n=abs(n); if( n<1, 0, direuler( p=2, n, 1 / (1 - X)^2)[n])}; /* Michael Somos, Apr 27 2003 */

{a(n)=polcoeff(sum(m=1, n+1, sumdiv(m, d, (-log(1-x^(m/d) +x*O(x^n) ))^d/d!)), n)} \\ Paul D. Hanna, Aug 21 2014

from sympy import divisor_count
for n in range(1, 20): print(divisor_count(n), end=', ') # Stefano Spezia, Nov 05 2018

[sigma(n, 0) for n in range(1, 105)]  # Zerinvary Lajos, Jun 04 2009

If n is written as 2^z*3^y*5^x*7^w*11^v*... then a(n)=(z+1)*(y+1)*(x+1)*(w+1)*(v+1)*...
a(n) = 2 iff n is prime.
G.f.: Sum_{n >= 1} a(n) x^n = Sum_{k>0} x^k/(1-x^k). This is usually called THE Lambert series (see Knopp, Titchmarsh).
a(n) = A083888(n) + A083889(n) + A083890(n) + A083891(n) + A083892(n) + A083893(n) + A083894(n) + A083895(n) + A083896(n).
a(n) = A083910(n) + A083911(n) + A083912(n) + A083913(n) + A083914(n) + A083915(n) + A083916(n) + A083917(n) + A083918(n) + A083919(n).
Multiplicative with a(p^e) = e+1. - David W. Wilson, Aug 01 2001
a(n) <= 2 sqrt(n) [see Mitrinovich, p. 39, also A046522].
a(n) is odd iff n is a square. - Reinhard Zumkeller, Dec 29 2001
a(n) = Sum_{k=1..n} f(k, n) where f(k, n) = 1 if k divides n, 0 otherwise (Mobius transform of A000012). Equivalently, f(k, n) = (1/k)*Sum_{l=1..k} z(k, l)^n with z(k, l) the k-th roots of unity. - Ralf Stephan, Dec 25 2002
G.f.: Sum_{k>0} ((-1)^(k+1) * x^(k * (k + 1)/2) / ((1 - x^k) * Product_{i=1..k} (1 - x^i))). - Michael Somos, Apr 27 2003
a(n) = n - Sum_{k=1..n} (ceiling(n/k) - floor(n/k)). - Benoit Cloitre, May 11 2003
a(n) = A032741(n) + 1 = A062011(n)/2 = A054519(n) - A054519(n-1) = A006218(n) - A006218(n-1) = 1 + Sum_{k=1..n-1} A051950(k+1). - Ralf Stephan, Mar 26 2004
G.f.: Sum_{k>0} x^(k^2)*(1+x^k)/(1-x^k). Dirichlet g.f.: zeta(s)^2. - Michael Somos, Apr 05 2003
Sequence = M*V where M = A129372 as an infinite lower triangular matrix and V = ruler sequence A001511 as a vector: [1, 2, 1, 3, 1, 2, 1, 4, ...]. - Gary W. Adamson, Apr 15 2007
Sequence = M*V, where M = A115361 is an infinite lower triangular matrix and V = A001227, the number of odd divisors of n, is a vector: [1, 1, 2, 1, 2, 2, 2, ...]. - Gary W. Adamson, Apr 15 2007
Row sums of triangle A051731. - Gary W. Adamson, Nov 02 2007
Sum_{n>0} a(n)/(n^n) = Sum_{n>0, m>0} 1/(n*m). - Gerald McGarvey, Dec 15 2007
Logarithmic g.f.: Sum_{n>=1} a(n)/n * x^n = -log( Product_{n>=1} (1-x^n)^(1/n) ). - Joerg Arndt, May 03 2008
a(n) = Sum_{k=1..n} (floor(n/k) - floor((n-1)/k)). - Enrique Pérez Herrero, Aug 27 2009
a(s) = 2^omega(s), if s > 1 is a squarefree number (A005117) and omega(s) is: A001221. - Enrique Pérez Herrero, Sep 08 2009
a(n) = A048691(n) - A055205(n). - Reinhard Zumkeller, Dec 08 2009
For n > 1, a(n) = 2 + Sum_{k=2..n-1} floor((cos(Pi*n/k))^2). And floor((cos(Pi*n/k))^2) = floor(1/4 * e^(-(2*i*Pi*n)/k) + 1/4 * e^((2*i*Pi*n)/k) + 1/2). - Eric Desbiaux, Mar 09 2010, corrected Apr 16 2011
a(n) = 1 + Sum_{k=1..n} (floor(2^n/(2^k-1)) mod 2) for every n. - Fabio Civolani (civox(AT)tiscali.it), Mar 12 2010
From Vladimir Shevelev, May 22 2010: (Start)
(Sum_{d|n} a(d))^2 = Sum_{d|n} a(d)^3 (J. Liouville).
Sum_{d|n} A008836(d)*a(d)^2 = A008836(n)*Sum_{d|n} a(d). (End)
a(n) = sigma_0(n) = 1 + Sum_{m>=2} Sum_{r>=1} (1/m^(r+1))*Sum_{j=1..m-1} Sum_{k=0..m^(r+1)-1} e^(2*k*Pi*i*(n+(m-j)*m^r)/m^(r+1)). - A. Neves, Oct 04 2010
a(n) = 2*A038548(n) - A010052(n). - Reinhard Zumkeller, Mar 08 2013
Sum_{n>=1} a(n)*q^n = (log(1-q) + psi_q(1)) / log(q), where psi_q(z) is the q-digamma function. - Vladimir Reshetnikov, Apr 23 2013
a(n) = Product_{k = 1..A001221(n)} (A124010(n,k) + 1). - Reinhard Zumkeller, Jul 12 2013
a(n) = Sum_{k=1..n} A238133(k)*A000041(n-k). - Mircea Merca, Feb 18 2013
G.f.: Sum_{k>=1} Sum_{j>=1} x^(j*k). - Mats Granvik, Jun 15 2013
The formula above is obtained by expanding the Lambert series Sum_{k>=1} x^k/(1-x^k). - Joerg Arndt, Mar 12 2014
G.f.: Sum_{n>=1} Sum_{d|n} ( -log(1 - x^(n/d)) )^d / d!. - Paul D. Hanna, Aug 21 2014
2*Pi*a(n) = Sum_{m=1..n} Integral_{x=0..2*Pi} r^(m-n)( cos((m-n)*x)-r^m cos(n*x) )/( 1+r^(2*m)-2r^m cos(m*x) )dx, 0 < r < 1 a free parameter. This formula is obtained as the sum of the residues of the Lambert series Sum_{k>=1} x^k/(1-x^k). - Seiichi Kirikami, Oct 22 2015
a(n) = A091220(A091202(n)) = A106737(A156552(n)). - Antti Karttunen, circa 2004 & Mar 06 2017
a(n) = A034296(n) - A237665(n+1) [Wang, Fokkink, Fokkink]. - George Beck, May 06 2017
G.f.: 2*x/(1-x) - Sum_{k>0} x^k*(1-2*x^k)/(1-x^k). - Mamuka Jibladze, Aug 29 2018
a(n) = Sum_{k=1..n} 1/phi(n / gcd(n, k)). - Daniel Suteu, Nov 05 2018
a(k*n) = a(n)*(f(k,n)+2)/(f(k,n)+1), where f(k,n) is the exponent of the highest power of k dividing n and k is prime. - Gary Detlefs, Feb 08 2019
a(n) = 2*log(p(n))/log(n), n > 1, where p(n)= the product of the factors of n = A007955(n). - Gary Detlefs, Feb 15 2019
a(n) = (1/n) * Sum_{k=1..n} sigma(gcd(n,k)), where sigma(n) = sum of divisors of n. - Orges Leka, May 09 2019
a(n) = A001227(n)*(A007814(n) + 1) = A001227(n)*A001511(n). - Ivan N. Ianakiev, Nov 14 2019
From Richard L. Ollerton, May 11 2021: (Start)
a(n) = A038040(n) / n = (1/n)*Sum_{d|n} phi(d)*sigma(n/d), where phi = A000010 and sigma = A000203.
a(n) = (1/n)*Sum_{k=1..n} phi(gcd(n,k))*sigma(n/gcd(n,k))/phi(n/gcd(n,k)). (End)
From Ridouane Oudra, Nov 12 2021: (Start)
a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*cos(2*k*n*Pi/j);
a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*e^(2*k*n*Pi*i/j), where i^2=-1. (End)

Incorrect formula deleted by Ridouane Oudra, Oct 28 2021

A020639 Lpf(n): least prime dividing n (when n > 1); a(1) = 1. Or, smallest prime factor of n, or smallest prime divisor of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 73, 2, 3, 2, 7, 2, 79, 2, 3, 2, 83, 2, 5, 2, 3, 2, 89, 2, 7, 2, 3, 2, 5, 2, 97

Offset: 1

David W. Wilson

Also, the largest number of distinct integers such that all their pairwise differences are coprime to n. - Max Alekseyev, Mar 17 2006
The unit 1 is not a prime number (although it has been considered so in the past). 1 is the empty product of prime numbers, thus 1 has no least prime factor. - Daniel Forgues, Jul 05 2011
a(n) = least m > 0 for which n! + m and n - m are not relatively prime. - Clark Kimberling, Jul 21 2012
For n > 1, a(n) = the smallest k > 1 that divides n. - Antti Karttunen, Feb 01 2014
For n > 1, records are at prime indices. - Zak Seidov, Apr 29 2015
The initials "lpf" might be mistaken for "largest prime factor" (A009190), using "spf" for "smallest prime factor" would avoid this. - M. F. Hasler, Jul 29 2015
n = 89 is the first index > 1 for which a(n) differs from the smallest k > 1 such that (2^k + n - 2)/k is an integer. - M. F. Hasler, Aug 11 2015
From Stanislav Sykora, Jul 29 2017: (Start)
For n > 1, a(n) is also the smallest k, 1 < k <= n, for which the binomial(n,k) is not divisible by n.
Proof: (A) When k and n are relatively prime then binomial(n,k) is divisible by n because k*binomial(n,k) = n*binomial(n-1,k-1). (B) When gcd(n,k) > 1, one of its prime factors is the smallest; let us denote it p, p <= k, and consider the binomial(n,p) = (1/p!)*Product_{i=0..p-1} (n-i). Since p is a divisor of n, it cannot be a divisor of any of the remaining numerator factors. It follows that, denoting as e the largest e > 0 such that p^e|n, the numerator is divisible by p^e but not by p^(e+1). Hence, the binomial is divisible by p^(e-1) but not by p^e and therefore not divisible by n. Applying (A), (B) to all considered values of k completes the proof. (End)
From Bob Selcoe, Oct 11 2017, edited by M. F. Hasler, Nov 06 2017: (Start)
a(n) = prime(j) when n == J (mod A002110(j)), n, j >= 1, where J is the set of numbers <= A002110(j) with smallest prime factor = prime(j). The number of terms in J is A005867(j-1). So:
a(n) = 2 when n == 0 (mod 2);
a(n) = 3 when n == 3 (mod 6);
a(n) = 5 when n == 5 or 25 (mod 30);
a(n) = 7 when n == 7, 49, 77, 91, 119, 133, 161 or 203 (mod 210);
etc. (End)
For n > 1, a(n) is the leftmost term, other than 0 or 1, in the n-th row of A127093. - Davis Smith, Mar 05 2019

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (terms 1..10000 from T. D. Noe)
A. E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Copy included with the permission of the author.]
OEIS Wiki, Least prime factor of n
David Singmaster, Letter to N. J. A. Sloane, Oct 3 1982.
Eric Weisstein's World of Mathematics, Least Prime Factor
Index entries for "core" sequences

Cf. A090368 (bisection).
Cf. A000040, A009190, A006530, A034684, A028233, A034699, A053585.
See also A032742, A055396, A068319, A088377, A007978, A053669, A117818.
Cf. A046669 (partial sums), A072486 (partial products).
Cf. A002110, A005867.
Cf. A127093.

a020639 n = spf a000040_list where
  spf (p:ps) | n < p^2      = n
             | mod n p == 0 = p
             | otherwise    = spf ps
-- Reinhard Zumkeller, Jul 13 2011

A020639 := proc(n) if n = 1 then 1; else min(op(numtheory[factorset](n))) ; end if; end proc: seq(A020639(n),n=1..20) ; # R. J. Mathar, Oct 25 2010

f[n_]:=FactorInteger[n][[1,1]]; Join[{1}, Array[f,120,2]]  (* Robert G. Wilson v, Apr 06 2011 *)
Join[{1}, Table[If[EvenQ[n], 2, FactorInteger[n][[1,1]]], {n, 2, 120}]] (* Zak Seidov, Nov 17 2013 *)
Riffle[Join[{1},Table[FactorInteger[n][[1,1]],{n,3,101,2}]],2] (* Harvey P. Dale, Dec 16 2021 *)

A020639(n) = { vecmin(factor(n)[,1]) } \\ [Will yield an error for n = 1.] - R. J. Mathar, Mar 02 2012

A020639(n)=if(n>1, if(n>n=factor(n,0)[1,1], n, factor(n)[1,1]), 1) \\ Avoids complete factorization if possible. Often the smallest prime factor can be found quickly even if it is larger than primelimit. If factoring takes too long for large n, use debugging level >= 3 (\g3) to display the smallest factor as soon as it is found. - M. F. Hasler, Jul 29 2015

from sympy import factorint
def a(n): return 1 if n == 1 else min(factorint(n))
print([a(n) for n in range(1, 98)]) # Michael S. Branicky, Dec 09 2021

def A020639_list(n) : return [1] + [prime_divisors(n)[0] for n in (2..n)]
A020639_list(97) # Peter Luschny, Jul 16 2012

[trial_division(n) for n in (1..100)] # Giuseppe Coppoletta, May 25 2016

(define (A020639 n) (if (< n 2) n (let loop ((k 2)) (cond ((zero? (modulo n k)) k) (else (loop (+ 1 k))))))) ;; Antti Karttunen, Feb 01 2014

A014673(n) = a(A032742(n)); A115561(n) = a(A054576(n)). - Reinhard Zumkeller, Mar 10 2006
A028233(n) = a(n)^A067029(n). - Reinhard Zumkeller, May 13 2006
a(n) = A027746(n,1) = A027748(n,1). - Reinhard Zumkeller, Aug 27 2011
For n > 1: a(n) = A240694(n,2). - Reinhard Zumkeller, Apr 10 2014
a(n) = A000040(A055396(n)) = n / A032742(n). - Antti Karttunen, Mar 07 2017
a(n) has average order n/(2 log n) [Brouwer] - N. J. A. Sloane, Sep 03 2017

Deleted wrong comment from M. Lagneau in 2012, following an observation by Gionata Neri. - M. F. Hasler, Aug 11 2015
Edited by M. F. Hasler, Nov 06 2017
Expanded definition to make this easier to find. - N. J. A. Sloane, Sep 21 2020

A027750 Triangle read by rows in which row n lists the divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 6, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 4, 7, 14, 28, 1, 29

Offset: 1

N. J. A. Sloane

Or, in the list of natural numbers (A000027), replace n with its divisors.
This gives the first elements of the ordered pairs (a,b) a >= 1, b >= 1 ordered by their product ab.
Also, row n lists the largest parts of the partitions of n whose parts are not distinct. - Omar E. Pol, Sep 17 2008
Concatenation of n-th row gives A037278(n). - Reinhard Zumkeller, Aug 07 2011
{A210208(n,k): k=1..A073093(n)} subset of {T(n,k): k=1..A000005(n)} for all n. - Reinhard Zumkeller, Mar 18 2012
Row sums give A000203. Right border gives A000027. - Omar E. Pol, Jul 29 2012
Indices of records are in A006218. - Irina Gerasimova, Feb 27 2013
The number of primes in the n-th row is omega(n) = A001221(n). - Michel Marcus, Oct 21 2015
The row polynomials P(n,x) = Sum_{k=1..A000005(n)} T(n,k)*x^k with composite n which are irreducible over the integers are given in A292226. - Wolfdieter Lang, Nov 09 2017
T(n,k) is also the number of parts in the k-th partition of n into equal parts (see example). - Omar E. Pol, Nov 20 2019
Let there be an infinite number of tiles, each labeled with a positive integer m, initially placed on square m of an infinite 1D board. At step n, the leftmost unblocked tile (i.e., the top tile of the leftmost nonempty stack) moves forward exactly m squares, where m is its label. Tiles that land on the same square form a stack, and only the top tile of any stack may move. This sequence records the label m of the tile that moves at step n. - Ali Sada, May 23 2025
All divisors of a positive integer n form a finite set. Extending divisibility to n = 0 by using the definition (k|n <=> exists m such that m*k = n) makes the set of divisors infinite, suggesting the definition was not intended for zero, as arithmetic functions typically apply to n >= 1. So to preserve a core property when generalizing (cardinality), one can define divisors of n >= 0 as the fixed points of the greatest common divisor on the set [n] = {0, 1, 2, ..., n}. By this definition, the divisors of 0 are {0}, since 0|0 and gcd(0, 0) = 0. This definition is not circular because the gcd can be effectively calculated using the Euclidean algorithm. (Cf. links.) - Peter Luschny, Jun 02 2025

			Triangle begins:
  1;
  1, 2;
  1, 3;
  1, 2, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 4, 8;
  1, 3, 9;
  1, 2, 5, 10;
  1, 11;
  1, 2, 3, 4, 6, 12;
  ...
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the number of parts are [1, 2, 3, 6] respectively, the same as the divisors of 6. - _Omar E. Pol_, Nov 20 2019

Franklin T. Adams-Watters, Rows 1..1000, flattened
Franklin T. Adams-Watters, Rows 1..10000
Peter Luschny, A Python program using gcd for n >= 0.
Omar E. Pol, Illustration of initial terms, (2009).
Eric Weisstein's World of Mathematics, Divisor
Wikipedia, Table of divisors
Index entries for sequences related to divisors of numbers

Cf. A000005 (row length), A001221, A027749, A027751, A056534, A056538, A127093, A135010, A161700, A163280, A240698 (partial sums of rows), A240694 (partial products of rows), A247795 (parities), A292226, A244051.

a027750 n k = a027750_row n !! (k-1)
a027750_row n = filter ((== 0) . (mod n)) [1..n]
a027750_tabf = map a027750_row [1..]
-- Reinhard Zumkeller, Jan 15 2011, Oct 21 2010

Magma
```
[Divisors(n) : n in [1..20]];
```

seq(op(numtheory:-divisors(a)), a = 1 .. 20) # Matt C. Anderson, May 15 2017

Flatten[ Table[ Flatten [ Divisors[ n ] ], {n, 1, 30} ] ]

v=List();for(n=1,20,fordiv(n,d,listput(v,d)));Vec(v) \\ Charles R Greathouse IV, Apr 28 2011

from sympy import divisors
for n in range(1, 16):
    print(divisors(n)) # Indranil Ghosh, Mar 30 2017

a(A006218(n-1) + k) = k-divisor of n, 1 <= k <= A000005(n). - Reinhard Zumkeller, May 10 2006

T(n,k) = n / A056538(n,k) = A056538(n,n-k+1), 1 <= k <= A000005(n). - Reinhard Zumkeller, Sep 28 2014

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A001157 a(n) = sigma_2(n): sum of squares of divisors of n.

Original entry on oeis.org

1, 5, 10, 21, 26, 50, 50, 85, 91, 130, 122, 210, 170, 250, 260, 341, 290, 455, 362, 546, 500, 610, 530, 850, 651, 850, 820, 1050, 842, 1300, 962, 1365, 1220, 1450, 1300, 1911, 1370, 1810, 1700, 2210, 1682, 2500, 1850, 2562, 2366, 2650, 2210, 3410, 2451, 3255

Offset: 1

N. J. A. Sloane, R. K. Guy

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
sigma_2(n) is the sum of the squares of the divisors of n.
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001.
Row sums of triangles A134575 and A134559. - Gary W. Adamson, Nov 02 2007
Also sum of square divisors of n^2. - Michel Marcus, Jan 14 2014
Conjecture: For each k = 2,3,..., all the rational numbers sigma_k(n)/n^k = Sum_{d|n} 1/d^k (n = 1,2,3,...) have pairwise distinct fractional parts. - Zhi-Wei Sun, Oct 15 2015
5 is the only prime entry in the sequence. - Drake Thomas, Dec 18 2016
4*a(n) = sum of squares of even divisors of 2*n. - Wolfdieter Lang, Jan 07 2017

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 827.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 11.
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table I. The entry 53 should be 50. - N. J. A. Sloane, May 21 2014
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See (2.3).
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for sequences related to sigma(n)
Index entries for "core" sequences

Cf. A000005, A000203, A001158, A001159, A053807, A064602, A127093, A134841, A048250.
Cf. also A192794, A082063 (gcd(a(n),n) and its largest prime factor); A179931, A192795 (gcd(a(n),A000203(n)) and largest prime factor).
Main diagonal of the array in A242639.
Cf. A333972 (Dgf at s=4).

a001157 n = s n 1 1 a000040_list where
   s 1 1 y _          = y
   s m x y ps'@(p:ps)
     | m `mod` p == 0 = s (m `div` p) (x * p^2) y ps'
     | x > 1          = s m 1 (y * (x * p^2 - 1) `div` (p^2 - 1)) ps
     | otherwise      = s m 1 y ps
-- Reinhard Zumkeller, Jul 10 2011

[DivisorSigma(2,n): n in [1..50]]; // Bruno Berselli, Apr 10 2013

with(numtheory); A001157 := n->sigma[2](n); [seq(sigma[2](n), n=1..100)];

Table[DivisorSigma[2, n], {n, 1, 50}] (* Stefan Steinerberger, Mar 24 2006 *)
DivisorSigma[2,Range[50]] (* Harvey P. Dale, Aug 22 2016 *)

makelist(divsum(n,2),n,1,20); /* Emanuele Munarini, Mar 26 2011 */

PARI
```
a(n)=if(n<1,0,sigma(n,2))
```

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1-p^2*X))[n])

a(n)=if(n<1,0,n*polcoeff(sum(k=1,n,x^k/(x^k-1)^2/k,x*O(x^n)),n)) /* Michael Somos, Jan 29 2005 */

N=99; q='q+O('q^N); Vec(sum(n=1,N,n^2*q^n/(1-q^n)))  /* Joerg Arndt, Feb 04 2011 */

a(n) = sumdiv(n^2, d, issquare(d)*d); \\ Michel Marcus, Jan 14 2014

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 2)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Jan 05 2021

from math import prod
from sympy import factorint
def a(n): return prod((p**(2*e+2)-1)//(p**2-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Feb 25 2024

[sigma(n,2)for n in range(1,51)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>0} k^2 x^k/(1-x^k). Dirichlet g.f.: zeta(s)*zeta(s-2). - Michael Somos, Apr 05 2003
Multiplicative with a(p^e) = (p^(2e+2)-1)/(p^2-1). - David W. Wilson, Aug 01 2001
G.f. for sigma_k(n): Sum_{m>0} m^k*x^m/(1-x^m). - Vladeta Jovovic, Oct 18 2002
L.g.f.: -log(Product_{j>=1} (1-x^j)^j) = Sum_{n>=1} a(n)/n*x^n. - Joerg Arndt, Feb 04 2011
Equals A127093 * [1, 2, 3, ...]. - Gary W. Adamson, May 10 2007
Equals A051731 * [1, 4, 9, 16, 25, ...]. A051731 * [1/1, 1/2, 1/3, 1/4, ...] = [1/1, 5/4, 10/9, 21/16, 26/25, ...]. - Gary W. Adamson, Nov 02 2007
Row sums of triangle A134841. - Gary W. Adamson, Nov 12 2007
a(n) = A035316(n^2). - Michel Marcus, Jan 14 2014
Conjecture: a(n) = sigma(n^2*rad(n))/sigma(rad(n)), where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 20 2017
G.f.: Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
a(n) = a(n/4) + A050461(n) + A076577(n/2) + A050465(n) where A(.) are zero for non-integer arguments. - R. J. Mathar, May 25 2020
Sum_{k>=1} 1/a(k) = A109694 = 1.53781289182725616253866100273826833091936004947322354929617689659426330445... - Vaclav Kotesovec, Sep 26 2020
G.f.: Sum_{n >= 1} q^(n^2)*(n^2 - ((n-1)^2 - 2)*q^n - ((n+1)^2 - 2)*q^(2*n) + n^2*q^(3*n))/(1 - q^n)^3 - apply the operator x*d/dx twice to equation 5 in Arndt and set x = 1. - Peter Bala, Jan 21 2021
From Vaclav Kotesovec, Aug 07 2022: (Start)
Sum_{k=1..n} a(k) = A064602(n) ~ zeta(3) * n^3 / 3.
Sum_{k=1..n} (-1)^k * a(k) ~ zeta(3) * n^3 / 24. (End)
a(n) = Sum_{1 <= i, j <= n} tau(gcd(i, j, n)) = Sum_{d divides n} tau(d) * J_2(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 22 2024

A001158 sigma_3(n): sum of cubes of divisors of n.

Original entry on oeis.org

1, 9, 28, 73, 126, 252, 344, 585, 757, 1134, 1332, 2044, 2198, 3096, 3528, 4681, 4914, 6813, 6860, 9198, 9632, 11988, 12168, 16380, 15751, 19782, 20440, 25112, 24390, 31752, 29792, 37449, 37296, 44226, 43344, 55261, 50654, 61740, 61544, 73710, 68922, 86688

Offset: 1

N. J. A. Sloane, R. K. Guy

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6..24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
Also the eigenvalues of the Hecke operator T_n for the entire modular normalized Eisenstein form E_4(z) (see A004009): T_n E_4 = a(n) E_4, n >= 1. For the Hecke operator T_n and eigenforms see, e.g., the Koecher-Krieg reference, p. 207, eq. (5) and p. 211, section 4, or the Apostol reference p. 120, eq. (13) and pp. 129 - 133. - Wolfdieter Lang, Jan 28 2016

			G.f. = x + 9*x^2 + 28*x^3 + 73*x^4 + 126*x^5 + 252*x^6 + 344*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 120, 129 - 133.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 166.
Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 207, 211.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_4(z).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
D. B. Lahiri, Some arithmetical identities for Ramanujan's and divisor functions, Bulletin of the Australian Mathematical Society, Volume 1, Issue 3 December 1969, pp. 307-314. See Theorem 2 p. 308.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Divisor Function.
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A001157.
Cf. A051731, A127093.
Cf. A027748, A124010.
Cf. A004009, A064603 (partial sums).

a001158 n = product $ zipWith (\p e -> (p^(3*e + 3) - 1) `div` (p^3 - 1))
                      (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jun 30 2013

[DivisorSigma(3,n): n in [1..40]]; // Bruno Berselli, Apr 10 2013

seq(numtheory:-sigma[3](n),n=1..100); # Robert Israel, Feb 05 2016

Table[DivisorSigma[3,n],{n,100}] (* corrected by T. D. Noe, Mar 22 2009 *)

makelist(divsum(n,3),n,1,100); /* Emanuele Munarini, Mar 26 2011 */

N=99; q='q+O('q^N);
Vec(sum(n=1,N,n^3*q^n/(1-q^n))) /* Joerg Arndt, Feb 04 2011 */

{a(n) = if( n<1, 0, sumdiv(n, d, d^3))}; /* Michael Somos, Jan 07 2017 */

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 3)
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Jan 09 2021

[sigma(n, 3) for n in range(1, 40)]  # Zerinvary Lajos, Jun 04 2009

Multiplicative with a(p^e) = (p^(3e+3)-1)/(p^3-1). - David W. Wilson, Aug 01 2001
Dirichlet g.f. zeta(s)*zeta(s-3). - R. J. Mathar, Mar 04 2011
G.f.: sum(k>=1, k^3*x^k/(1-x^k)). - Benoit Cloitre, Apr 21 2003
Equals A051731 * [1, 8, 27, 64, 125, ...] = A127093 * [1, 4, 9, 16, 25, ...]. - Gary W. Adamson, Nov 02 2007
L.g.f.: -log(Product_{j>=1} (1-x^j)^(j^2)) = (1/1)*z^1 + (9/2)*z^2 + (28/3)*z^3 + (73/4)*z^4 + ... + (a(n)/n)*z^n + ... - Joerg Arndt, Feb 04 2011
a(n) = Sum{d|n} tau_{-2}^d*J_3(n/d), where tau_{-2} is A007427 and J_3 is A059376. - Enrique Pérez Herrero, Jan 19 2013
a(n) = A004009(n)/240. - Artur Jasinski, Sep 06 2016. See, e.g., Hardy, p. 166, (10.5.6), with Q = E_4, and with present offset 0. - Wolfdieter Lang, Jan 31 2017
8*a(n) = sum of cubes of even divisors of 2*n. - Wolfdieter Lang, Jan 07 2017
G.f.: Sum_{n >= 1} x^n*(1 + 4*x^n + x^(2*n))/(1 - x^n)^4. - Peter Bala, Jan 11 2021
Faster converging g.f.: Sum_{n >= 1} q^(n^2)*( n^3 + ((n + 1)^3 - 3*n^3)*q^n + (4 - 6*n^2)*q^(2*n) + (3*n^3 - (n - 1)^3)*q^(3*n) - n^3*q^(4*n) )/(1 - q^n)^4 - apply the operator x*d/dx three times to equation 5 in Arndt and then set x = 1. - Peter Bala, Jan 21 2021
a(n) = Sum_{1 <= i, j, k <= n} tau(gcd(i, j, k, n)) = Sum_{d divides n} tau(d)* J_3(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 22 2024

A038040 a(n) = n*d(n), where d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 4, 6, 12, 10, 24, 14, 32, 27, 40, 22, 72, 26, 56, 60, 80, 34, 108, 38, 120, 84, 88, 46, 192, 75, 104, 108, 168, 58, 240, 62, 192, 132, 136, 140, 324, 74, 152, 156, 320, 82, 336, 86, 264, 270, 184, 94, 480, 147, 300, 204, 312, 106, 432, 220, 448, 228, 232, 118

Offset: 1

Christian G. Bower

Dirichlet convolution of sigma(n) (A000203) with phi(n) (A000010). - Michael Somos, Jun 08 2000
Dirichlet convolution of f(n)=n with itself. See the Apostol reference for Dirichlet convolutions. - Wolfdieter Lang, Sep 09 2008
Sum of all parts of all partitions of n into equal parts. - Omar E. Pol, Jan 18 2013

			For n = 6 the partitions of 6 into equal parts are [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. The sum of all parts is 6 + 3 + 3 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 = 24 equalling 6 times the number of divisors of 6, so a(6) = 24. - _Omar E. Pol_, May 08 2021

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 29 ff.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 162.

T. D. Noe, Table of n, a(n) for n = 1..1000
J. Bourgain, S. V. Konyagin and I. E. Shparlinski, Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithms, Int. Math. Res. Notices, 2008 (2008), Art. ID rnn 090, 1-29.
Jean Bourgain, Sergei Konyagin and Igor Shparlinski. Distribution on elements of cosets of small subgroups and applications, arXiv:1103.0567 [math.NT], Mar 2 2011.
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147. [Broken link?]
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147.

Cf. A000005, A000010, A000203, A001620, A018804, A029935, A064987, A062952.
Cf. A127648, A127093, A127528.
Cf. A038044, A143127 (partial sums), A328722 (Dirichlet inverse).
Column 1 of A329323.

a038040 n = a000005 n * n  -- Reinhard Zumkeller, Jan 21 2014

with(numtheory): A038040 := n->tau(n)*n;

a[n_] := DivisorSigma[0, n]*n; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 03 2012 *)

n*numlib::tau (n)$ n=1..90 // Zerinvary Lajos, May 13 2008

a(n)=if(n<1,0,direuler(p=2,n,1/(1-p*X)^2)[n])

a(n)=if(n<1,0,polcoeff(sum(k=1,n,k*x^k/(x^k-1)^2,x*O(x^n)),n)) /* Michael Somos, Jan 29 2005 */

a(n) = n*numdiv(n); \\ Michel Marcus, Oct 24 2020

from sympy import divisor_count as d
def a(n): return n*d(n)
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Mar 15 2022

[n*sigma(n,0) for n in range(1, 60)] # Stefano Spezia, Jul 20 2025

Dirichlet g.f.: zeta(s-1)^2.
G.f.: Sum_{n>=1} n*x^n/(1-x^n)^2. - Vladeta Jovovic, Dec 30 2001
Sum_{k=1..n} sigma(gcd(n, k)). Multiplicative with a(p^e) = (e+1)*p^e. - Vladeta Jovovic, Oct 30 2001
Equals A127648 * A127093 * the harmonic series, [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, May 10 2007
Equals row sums of triangle A127528. - Gary W. Adamson, May 21 2007
a(n) = n*A000005(n) = A066186(n) - n*(A000041(n) - A000005(n)) = A066186(n) - n*A144300(n). - Omar E. Pol, Jan 18 2013
a(n) = A000203(n) * A240471(n) + A106315(n). - Reinhard Zumkeller, Apr 06 2014
L.g.f.: Sum_{k>=1} x^k/(1 - x^k) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 13 2017
a(n) = Sum_{d|n} A018804(d). - Amiram Eldar, Jun 23 2020
a(n) = Sum_{d|n} phi(d)*sigma(n/d). - Ridouane Oudra, Jan 21 2021
G.f.: Sum_{n >= 1} q^(n^2)*(n^2 + 2*n*q^n - n^2*q^(2*n))/(1 - q^n)^2. - Peter Bala, Jan 22 2021
a(n) = Sum_{k=1..n} sigma(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ x/sqrt(log x). That is, there are 0 < A < B such that Ax/sqrt(log x) < f(x) < Bx/sqrt(log x). - Charles R Greathouse IV, Mar 15 2022
Sum_{k=1..n} a(k) ~ n^2*log(n)/2 + (gamma - 1/4)*n^2, where gamma is Euler's constant (A001620). - Amiram Eldar, Oct 25 2022
Mobius transform of A060640. - R. J. Mathar, Feb 07 2023

A191898 Symmetric square array read by antidiagonals: T(n,1)=1, T(1,k)=1, T(n,k) = -Sum_{i=1..k-1} T(n-i,k) for n >= k, -Sum_{i=1..n-1} T(k-i,n) for n < k.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -2, 1, 1, -2, 1, 1, 1, -1, 1, -1, -4, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -6, -1, 1, -1, 1, -1, 1

Offset: 1

Mats Granvik, Jun 19 2011

Rows equal columns and are periodic. No zero elements are found (conjecture). The recurrence is related to the recurrence for the Mahonian numbers. The main diagonal is the Dirichlet inverse of the Euler totient function A023900 (conjecture). [The 2nd and 3rd formulas state that the conjecture is correct. R. J. Mathar, Sep 16 2017]
The sums from n=1 to infinity of T(n,k)/n converge to the Mangoldt function for column k (conjecture).
If gcd(n,k)=1 then T(n,k)=1 and if T(n,k)=1 then gcd(n,k)=1 (conjecture).
The Dirichlet generating functions for s > 1 for the columns appear to be (see A054535):
Zeta(s)*(1 + (Sum over all possible combinations of products of negative distinct prime factors of k, up to rearrangement, 1/((-1* first distinct prime factor)*(-1*second distinct prime factor)*(-1*third distinct prime factor * ...))^(s-1))).
Examples:
k=1: Zeta(s)
k=2: Zeta(s)*(1 - 1/2^(s-1))
k=3: Zeta(s)*(1 - 1/3^(s-1))
k=4: Zeta(s)*(1 - 1/2^(s-1))
k=5: Zeta(s)*(1 - 1/5^(s-1))
k=6: Zeta(s)*(1 - 1/2^(s-1) - 1/3^(s-1) + 1/6^(s-1))
k=7: Zeta(s)*(1 - 1/7^(s-1))
...
k=30: Zeta(s)*(1 - 1/2^(s-1) - 1/3^(s-1) - 1/5^(s-1) + 1/6^(s-1) + 1/10^(s-1) + 1/15^(s-1) - 1/30^(s-1))
...
(conjecture)
See triangle A142971 for negative distinct prime factors.
This could probably be checked by matrix multiplication.
The signs of the eigenvalues of this matrix are a rearrangement of the Mobius function A008683 (conjecture). The first few eigenvalues are:
{1.0000}
{-1.4142, 1.4142}
{-2.6554, 1.8662, -1.2108}
{-3.4393, 2.1004, -1.6611, 0}
{-4.7711, -3.3867, 2.5910, -1.4332, 0}
{-5.2439, -3.4641, 3.4641, 2.5169, -2.2730, 0}
The relation to Dirichlet characters for the entries in this matrix appears appears to be formulated in terms of the sequence A089026 which is equal to n if n is a prime, otherwise equal to 1. See Mathematica program below. [Mats Granvik, Nov 23 2013]
From Mats Granvik, Jun 19 2016: (Start)
Remark about the Dirichlet generating function for the whole matrix: Subtracting the first column (in the form of zeta(c)) of the matrix gives us the limit: lim_{c->1} zeta(s)*zeta(c)/zeta(c+s-1)-zeta(c) = -zeta'(s)/zeta(s) which is the classical Dirichlet generating function for the von Mangoldt function.
For n >= k, see A231425, this matrix has row sums equal to zero except for the first row:
1=1
1-1=0
1+1-2=0
1-1+1-1=0
1+1+1+1-4=0
...
log(A014963(n)) = Sum_{k>=1} A191898(n,k)/k, for n>1.
log(A014963(k)) = Sum_{n>=1} A191898(n,k)/n, for k>1.
log(A014963(n)) = limit of zeta(s)*(Sum_{d divides n} A008683(d)/d^(s-1)) as s->1, for n>1.
A008683(n) = Sum_{k=1..n} A191898(n,k)*exp(-i*2*Pi*k/n)/n.
A008683(n) = Sum_{n=1..k} A191898(n,k)*exp(-i*2*Pi*n/k)/k.
(End)
From Mats Granvik, Aug 09 2016: (Start)
The Dirichlet generating function for the matrix is zero for any pair c and s = 2 - c and any pair s and c = 2 - s  except at the pole c = 1 and s = 1 where it is indeterminate.
In the Mathematica program section, in the expression for the matrix as Dirichlets characters, the variables s and c can apparently be any pair of positive integers.
Limits related to the Dirichlet generating function for the matrix: Let s = ZetaZero(n), then lim_{c->1} zeta(s*c)/zeta(c+s-1) = ZetaZero(n). Let s = ZetaZero(n), then lim_{c->1} zeta(s*c)/zeta(c+s*c-1) = ZetaZero(n)/(1+ZetaZero(n)).
(End)

			Array starts:
n\k | 1    2    3    4    5    6    7    8    9   10
----+-----------------------------------------------------
1   | 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
2   | 1,  -1,   1,  -1,   1,  -1,   1,  -1,   1,  -1, ...
3   | 1,   1,  -2,   1,   1,  -2,   1,   1,  -2,   1, ...
4   | 1,  -1,   1,  -1,   1,  -1,   1,  -1,   1,  -1, ...
5   | 1,   1,   1,   1,  -4,   1,   1,   1,   1,  -4, ...
6   | 1,  -1,  -2,  -1,   1,   2,   1,  -1,  -2,  -1, ...
7   | 1,   1,   1,   1,   1,   1,  -6,   1,   1,   1, ...
8   | 1,  -1,   1,  -1,   1,  -1,   1,  -1,   1,  -1, ...
9   | 1,   1,  -2,   1,   1,  -2,   1,   1,  -2,   1, ...
10  | 1,  -1,   1,  -1,  -4,  -1,   1,  -1,   1,   4, ...

Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 antidiagonals of the array
Mats Granvik, Is this similarity to the Fourier transform of the von Mangoldt function real?
Mats Granvik, Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?
Mats Granvik, Primes approximated by eigenvalues?
Mats Granvik, Are the primes found as a subset in this sequence a(n)?
Mats Granvik, Will every eigenvalue in this type of matrix eventually be a common eigenvalue to infinitely many subsequent larger matrices of the same form?
Mats Granvik, How write Dirichlet character sums for the terms of the von Mangoldt function?
Mats Granvik, Do these series converge to the von Mangoldt function?
Mats Granvik, Is this sum equal to the Möbius function?
Mats Granvik, Can the Riemann hypothesis be relaxed to say that this matrix A consists of square roots?
Mats Granvik, Elementary proof of the prime number theorem?
Mats Granvik, Is this Dirichlet series generating function of the von Mangoldt function matrix correct?
Mats Granvik, Question about ratios of polynomials evaluated at x=1

Cf. A023900, A014963, A008302, A033999, A061347, A142971.
Cf. A007917, A199514, A260237. - Mats Granvik, Jun 19 2016
Cf. A343555, A343556.

T[ n_, k_] := T[ n, k] = Which[ n < 1 || k < 1, 0, n == 1 || k == 1, 1, k > n, T[k, n], n > k,T[k, Mod[n, k, 1]], True, -Sum[ T[n, i], {i, n - 1}]]; (* Michael Somos, Jul 18 2011 *)
(* Conjectured expression for the matrix as Dirichlet characters *) s = RandomInteger[{1, 3}]; c = RandomInteger[{1, 3}]; nn = 12; b = Table[Exp[MangoldtLambda[Divisors[n]]]^-MoebiusMu[Divisors[n]], {n, 1, nn^Max[s, c]}]; j = 1; MatrixForm[Table[Table[Product[(b[[n^s]][[m]]*DirichletCharacter[b[[n^s]][[m]], j, k^c] - (b[[n^s]][[m]] - 1)), {m, 1, Length[Divisors[n]]}], {n, 1, nn}], {k, 1, nn}]] (* Mats Granvik, Nov 23 2013 and Aug 09 2016 *)

{T(n, k) = if( n<1 || k<1, 0, n==1 || k==1, 1, k>n, T(k, n), kMichael Somos, Jul 18 2011 */

from sympy.core.cache import cacheit
@cacheit
def T(n, k): return 0 if n<1 or k<1 else 1 if n==1 or k==1 else T(k, n) if k>n else T(k, (n - 1)%k + 1) if n>k else -sum([T(n, i) for i in range(1, n)])
for n in range(1, 21): print([T(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Oct 23 2017

T(n,1)=1, T(1,k)=1, n>=k: -Sum_{i=1..k-1} T(n-i,k), n

T(n, n) = A023900(n). - Michael Somos, Jul 18 2011

T(n, k) = A023900(gcd(n,k)). - Mats Granvik, Jun 18 2012

Dirichlet generating function for sequence in the n-th row: zeta(s)*Sum_{ d divides n } mu(d)/d^(s-1). - Mats Granvik, Jun 18 2012 & Jun 19 2016

From Mats Granvik, Jun 19 2016: (Start)

Dirichlet generating function for the whole matrix: Sum_{k>=1} (Sum_{n>=1} T(n,k)/(n^c*k^s)) = Sum_{n>=1} (zeta(s)*Sum_{ d divides n } mu(d)/d^(s-1))/n^c = zeta(s)*zeta(c)/zeta( c + s - 1 ).

T(n,k) = A127093(n,k)^(1/2-i*a(k))*transpose(A008683(k)*(A127093(n,k)^(1/2+i*a(n)))) where a(x) is some real number. An example would be T(n,k) = A127093(n,k)^(zetazero(k))*transpose(A008683(k)*(A127093(n,k)^(zetazero(-k)))) but this is of course not special for only the zeta zeros.

Recurrence for a subset of A191898 that is a cross-directional variant of the recurrence in A051731: T(1,1)=1, T(1,2..k)=0, T(2..n,1)=0, n >= k: -Sum_{i=1..k-1} T(n-i,k) - T(n-i,k-1), n < k: -Sum_{i=1..n-1} T(k-i,n) - T(k-i,n-1). Notice that the identity matrix in linear algebra satisfies a similar recurrence:

T(1,1)=1, T(1,2..k)=0, T(2..n,1)=0, n >= k: -Sum_{i=1..n-1} T(n-i,k) - T(n-i,k-1), n < k: -Sum_{i=1..k-1} T(k-i,n) - T(k-i,n-1).

(End)

This array equals A051731*transpose(A143256). - Mats Granvik, Jul 22 2016

T(n,k) = sqrt(A143256(n,k))*transpose(sqrt(A143256(n,k))). - Mats Granvik, Aug 10 2018

Dirichlet generating function for absolute values: Sum_{k>=1} (Sum_{n>=1} abs(T(n,k))/(n^c*k^s)) = zeta(s)*zeta(c)*zeta(s + c - 1)/zeta(2*(s + c - 1))*Product_{k>=1} (1 - 2/(prime(k) + prime(k)^(s + c))). After Vaclav Kotesovec in A173557. - Mats Granvik, Apr 25 2021

A338156 Irregular triangle read by rows in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the divisors of (n - m + 1), with 1 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 5, 1, 2, 4, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 6, 1, 5, 1, 2, 4, 1, 2, 4, 1, 3, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 2, 3, 6, 1, 5, 1, 5, 1, 2, 4, 1, 2, 4, 1, 2, 4

Offset: 1

Omar E. Pol, Oct 14 2020

In other words: in row n replace every term of n-th row of A176206 with its divisors.
The terms in row n are also all parts of all partitions of n.
As in A336812 here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the correspondence between all divisors of all terms of the n-th row of A176206 and all parts of all partitions of n, with n >= 1. Both the mentionded divisors and the mentioned parts are the same numbers (see Example section). That is because all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.
For an equivalent table for all parts of the last section of the set of partitions of n see the subsequence A336812. The section is the smallest substructure of the set of partitions in which appears the correspondence divisor/part.
From Omar E. Pol, Aug 01 2021: (Start)
The terms of row n appears in the triangle A346741 ordered in accordance with the successive sections of the set of partitions of n.
The terms of row n in nonincreasing order give the n-th row of A302246.
The terms of row n in nondecreasing order give the n-th row of A302247.
For the connection with the tower described in A221529 see also A340035. (End)

			Triangle begins:
  [1];
  [1,2],   [1];
  [1,3],   [1,2],   [1],   [1];
  [1,2,4], [1,3],   [1,2], [1,2], [1],   [1],   [1];
  [1,5],   [1,2,4], [1,3], [1,3], [1,2], [1,2], [1,2], [1], [1], [1], [1], [1];
  ...
For n = 5 the 5th row of A176206 is [5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1] so replacing every term with its divisors we have the 5th row of this triangle.
Also, if the sequence is written as an irregular tetrahedron so the first six slices are:
  [1],
  -------
  [1, 2],
  [1],
  -------
  [1, 3],
  [1, 2],
  [1],
  [1];
  ----------
  [1, 2, 4],
  [1, 3],
  [1, 2],
  [1, 2],
  [1],
  [1],
  [1];
  ----------
  [1, 5],
  [1, 2, 4],
  [1, 3],
  [1, 3],
  [1, 2],
  [1, 2],
  [1, 2],
  [1],
  [1],
  [1],
  [1],
  [1];
.
The above slices appear in the lower zone of the following table which shows the correspondence between the mentioned divisors and all parts of all partitions of the positive integers.
The table is infinite. It is formed by three zones as follows:
The upper zone shows the partitions of every positive integer in colexicographic order (cf. A026792, A211992).
The lower zone shows the same numbers but arranged as divisors in accordance with the slices of the tetrahedron mentioned above.
Finally the middle zone shows the connection between the upper zone and the lower zone.
For every positive integer the numbers in the upper zone are the same numbers as in the lower zone.
.
|---|---------|-----|-------|---------|------------|---------------|
| n |         |  1  |   2   |    3    |      4     |       5       |
|---|---------|-----|-------|---------|------------|---------------|
| P |         |     |       |         |            |               |
| A |         |     |       |         |            |               |
| R |         |     |       |         |            |               |
| T |         |     |       |         |            |  5            |
| I |         |     |       |         |            |  3  2         |
| T |         |     |       |         |  4         |  4  1         |
| I |         |     |       |         |  2  2      |  2  2  1      |
| O |         |     |       |  3      |  3  1      |  3  1  1      |
| N |         |     |  2    |  2 1    |  2  1 1    |  2  1  1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1  1 1 1  |  1  1  1 1 1  |
----|---------|-----|-------|---------|------------|---------------|
.
|---|---------|-----|-------|---------|------------|---------------|
|   | A181187 |  1  |  3 1  |  6 2 1  | 12  5 2 1  | 20  8  4 2 1  |
|   |         |  |  |  |/|  |  |/|/|  |  |/ |/|/|  |  |/ | /|/|/|  |
| L | A066633 |  1  |  2 1  |  4 1 1  |  7  3 1 1  | 12  4  2 1 1  |
| I |         |  *  |  * *  |  * * *  |  *  * * *  |  *  *  * * *  |
| N | A002260 |  1  |  1 2  |  1 2 3  |  1  2 3 4  |  1  2  3 4 5  |
| K |         |  =  |  = =  |  = = =  |  =  = = =  |  =  =  = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7  6 3 4  | 12  8  6 4 5  |
|   |         |  |  |  |\|  |  |\|\|  |  |\ |\|\|  |  |\ |\ |\|\|  |
|   | A206561 |  1  |  4 2  |  9 5 3  | 20 13 7 4  | 35 23 15 9 5  |
|---|---------|-----|-------|---------|------------|---------------|
.
|---|---------|-----|-------|---------|------------|---------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1  2   4  |  1         5  |
|   |---------|-----|-------|---------|------------|---------------|
|   | A027750 |     |  1    |  1 2    |  1    3    |  1  2    4    |
|   |---------|-----|-------|---------|------------|---------------|
| D | A027750 |     |       |  1      |  1  2      |  1     3      |
| I | A027750 |     |       |  1      |  1  2      |  1     3      |
| V |---------|-----|-------|---------|------------|---------------|
| I | A027750 |     |       |         |  1         |  1  2         |
| S | A027750 |     |       |         |  1         |  1  2         |
| O | A027750 |     |       |         |  1         |  1  2         |
| R |---------|-----|-------|---------|------------|---------------|
| S | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|---|---------|-----|-------|---------|------------|---------------|
.
Note that every row in the lower zone lists A027750.
Also the lower zone for every positive integer can be constructed using the first n terms of the partition numbers. For example: for n = 5 we consider the first 5 terms of A000041 (that is [1, 1, 2, 3, 5]) then the 5th slice is formed by a block with the divisors of 5, one block with the divisors of 4, two blocks with the divisors of 3, three blocks with the divisors of 2, and five blocks with the divisors of 1.
Note that the lower zone is also in accordance with the tower (a polycube) described in A221529 in which its terraces are the symmetric representation of sigma starting from the top (cf. A237593) and the heights of the mentioned terraces are the partition numbers A000041 starting from the base.
The tower has the same volume (also the same number of cubes) equal to A066186(n) as a prism of partitions of size 1*n*A000041(n).
The above table shows the correspondence between the prism of partitions and its associated tower since the number of parts in all partitions of n is equal to A006128(n) equaling the number of divisors in the n-th slice of the lower table and equaling the same the number of terms in the n-th row of triangle. Also the sum of all parts of all partitions of n is equal to A066186(n) equaling the sum of all divisors in the n-th slice of the lower table and equaling the sum of the n-th row of triangle.

Paolo Xausa, Table of n, a(n) for n = 1..13185 (rows 1..19 of triangle, flattened)

Nonzero terms of A340031.
Row n has length A006128(n).
The sum of row n is A066186(n).
The product of row n is A007870(n).
Row n lists the first n rows of A336812 (a subsequence).
The number of parts k in row n is A066633(n,k).
The sum of all parts k in row n is A138785(n,k).
The number of parts >= k in row n is A181187(n,k).
The sum of all parts >= k in row n is A206561(n,k).
The number of parts <= k in row n is A210947(n,k).
The sum of all parts <= k in row n is A210948(n,k).
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A127093, A135010, A138121, A176206, A182703, A206437, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340035, A340061, A346741.

A338156[rowmax_]:=Table[Flatten[Table[ConstantArray[Divisors[n-m],PartitionsP[m]],{m,0,n-1}]],{n,rowmax}];
A338156[10] (* Generates 10 rows *) (* Paolo Xausa, Jan 12 2023 *)

A338156(rowmax)=vector(rowmax,n,concat(vector(n,m,concat(vector(numbpart(m-1),i,divisors(n-m+1))))));
A338156(10) \\ Generates 10 rows - Paolo Xausa, Feb 17 2023

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A143111 Triangle read by rows, T(n,k) = largest proper divisor of A127093(n,k) where (largest proper divisor)(n) = A032742(n) if n>0 and 0 if n=0.

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A000203 a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).

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A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.

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A020639 Lpf(n): least prime dividing n (when n > 1); a(1) = 1. Or, smallest prime factor of n, or smallest prime divisor of n.

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