A001227 -id:A001227 - 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.

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.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 116ff.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 407.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 2nd formula.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 141, 166.
H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
Ross Honsberger, "Mathematical Gems, Number One," The Dolciani Mathematical Expositions, Published and Distributed by The Mathematical Association of America, page 116.
Kanold, Hans Joachim, Kreisteilungspolynome und ungerade vollkommene Zahlen. (German), Ber. Math.-Tagung Tübingen 1946, (1947). pp. 84-87.
M. Krasner, Le nombre des surcorps primitifs d'un degré donné et le nombre des surcorps métagaloisiens d'un degré donné d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Académie des Sciences, Paris 254, 255, 1962.
A. Lubotzky, Counting subgroups of finite index, Proceedings of the St. Andrews/Galway 93 group theory meeting, Th. 2.1. LMS Lecture Notes Series no. 212 Cambridge University Press 1995.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.1, page 77.
G. Pólya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ Press 1954, page 92.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 91, 395.
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)
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].
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.
D. Christopher and T. Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), #15.11.5.
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.
L. Euler, Discovery of a most extraordinary law of numbers, relating to the sum of their divisors
L. Euler, Observatio de summis divisorum
L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009.
J. A. Ewell, Recurrences for the sum of divisors, Proc. Amer. Math. Soc. 64 (2) 1977.
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.
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]
M. J. Grady, A group theoretic approach to a famous partition formula, Amer. Math. Monthly, 112 (No. 7, 2005), 645-651.
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
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 3.
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

cp's OEIS Frontend

A318454 Denominators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

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A324593 a(n) is the smallest number k such that n consecutive integers starting at k have the same number of odd divisors (A001227).

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A366797 Number of odd divisors permuted by A163511: a(n) = A001227(A163511(n)).

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A209445 a(n) = Pell(n)*A001227(n) for n >= 1, where A001227(n) is the number of odd divisors of n.

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A349913 Sum of A001227 (the number of odd divisors function) and its Dirichlet inverse.

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A309153 a(n) = A000203(n)*A001227(n).

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A348623 a(1) = 1; for n > 1 a(n) = a(n-1) + A001227(a(n-1)).

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A371533 Numbers k such that A001227(2*k) = A001222(2*k).

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A371533 Numbers k such that A001227(2k) = A001222(2k).