A006128 -id:A006128 - 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)
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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.
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

A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, 21, 46, 1, 66, 17, 64, 23, 32, 1, 108, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 123, 1, 40, 49, 64, 19, 90, 1, 106

Offset: 1

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

Also total number of parts in all partitions of n into equal parts that do not contain 1 as a part. - Omar E. Pol, Jan 16 2013
Related concepts: If a(n) < n, n is said to be deficient, if a(n) > n, n is abundant, and if a(n) = n, n is perfect. If there is a cycle of length 2, so that a(n) = b and a(b) = n, b and n are said to be amicable. If there is a longer cycle, the numbers in the cycle are said to be sociable. See examples. - Juhani Heino, Jul 17 2017
Sum of the smallest parts in the partitions of n into two parts such that the smallest part divides the largest. - Wesley Ivan Hurt, Dec 22 2017
a(n) is also the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that do not contain k as a part (the comment dated Jan 16 2013 is the case for k = 1). - Omar E. Pol, Nov 23 2019
Fixed points are in A000396. - Alois P. Heinz, Mar 10 2024

			x^2 + x^3 + 3*x^4 + x^5 + 6*x^6 + x^7 + 7*x^8 + 4*x^9 + 8*x^10 + x^11 + ...
For n = 44, sum of divisors of n = sigma(n) = 84; so a(44) = 84-44 = 40.
Related concepts: (Start)
From 1 to 17, all n are deficient, except 6 and 12 seen below. See A005100.
Abundant numbers: a(12) = 16, a(18) = 21. See A005101.
Perfect numbers: a(6) = 6, a(28) = 28. See A000396.
Amicable numbers: a(220) = 284, a(284) = 220. See A259180.
Sociable numbers: 12496 -> 14288 -> 15472 -> 14536 -> 14264 -> 12496. See A122726. (End)
For n = 10 the sum of the divisors of 10 that are less than 10 is 1 + 2 + 5 = 8. On the other hand, the partitions of 10 into equal parts that do not contain 1 as a part are [10], [5,5], [2,2,2,2,2], there are 8 parts, so a(10) = 8. - _Omar E. Pol_, Nov 24 2019

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.
George E. Andrews, Number Theory. New York: Dover, 1994; Pages 1, 75-92; p. 92 #15: Sigma(n) / d(n) >= n^(1/2).
Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.
H. J. J. te Riele, Perfect numbers and aliquot sequences, pp. 77-94 in J. van de Lune, ed., Studieweek "Getaltheorie en Computers", published by Math. Centrum, Amsterdam, Sept. 1980.
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, page 91.

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].
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Henry Bottomley, Illustration of initial terms
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences, Exper. Math. 29 (2020), no. 4, 414-425; arXiv:2110.14136, Oct. 2021 [math.NT].
Don Coppersmith, An answer to the problem of Don Saari, 1987.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
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., to appear (2014).
Primefan, Sums of Restricted Divisors for n=1 to 1000
F. Richman, Aliquot series: Abundant, deficient, perfect
Eric Weisstein's World of Mathematics, Restricted Divisor Function
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for "core" sequences

Least inverse: A070015, A359132.
Values taken: A078923, values not taken: A005114.
Records: A034090, A034091.
First differences: A053246, partial sums: A153485.
a(n) = n - A033879(n) = n + A033880(n). - Omar E. Pol, Dec 30 2013
Row sums of A141846 and of A176891. - Gary W. Adamson, May 02 2010
Row sums of A176079. - Mats Granvik, May 20 2012
Alternating row sums of A231347. - Omar E. Pol, Jan 02 2014
a(n) = sum (A027751(n,k): k = 1..A000005(n)-1). - Reinhard Zumkeller, Apr 05 2013
For n > 1: a(n) = A240698(n,A000005(n)-1). - Reinhard Zumkeller, Apr 10 2014
A134675(n) = A007434(n) + a(n). - Conjectured by John Mason and proved by Max Alekseyev, Jan 07 2015
Cf. A032741, A000203, A048050, A000593, A027750.
Cf. A051953, A051731.
Cf. A037020 (primes), A053868, A053869 (odd and even terms).
Cf. A048138 (number of occurrences), A238895, A238896 (record values thereof).
Cf. A007956 (products of proper divisors).
Cf. A005100, A005101, A000396, A259180, A122726 (related concepts).

a001065 n = a000203 n - n  -- Reinhard Zumkeller, Sep 15 2011

[SumOfDivisors(n)-n: n in [1..100]]; // Vincenzo Librandi, May 06 2015

A001065 := proc(n)
    numtheory[sigma](n)-n ;
end proc:
seq( A001065(n),n=1..100) ;

Table[ Plus @@ Select[ Divisors[ n ], #Zak Seidov, Sep 10 2009 *)
Table[DivisorSigma[1, n] - n, {n, 1, 80}] (* Jean-François Alcover, Apr 25 2013 *)
Array[Plus @@ Most@ Divisors@# &, 80] (* Robert G. Wilson v, Dec 24 2017 *)

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

{a(n) = if( n==0, 0, sigma(n) - n)} /* Michael Somos, Sep 20 2011 */

from sympy import divisor_sigma
def A001065(n): return divisor_sigma(n)-n # Chai Wah Wu, Nov 04 2022

[sigma(n, 1)-n for n in range(1, 81)] # Stefano Spezia, Jul 14 2025

G.f.: Sum_{k>0} k * x^(2*k)/(1 - x^k). - Michael Somos, Jul 05 2006
a(n) = sigma(n) - n = A000203(n) - n. - Lekraj Beedassy, Jun 02 2005
a(n) = A155085(-n). - Michael Somos, Sep 20 2011
Equals inverse Mobius transform of A051953 = A051731 * A051953. Example: a(6) = 6 = (1, 1, 1, 0, 0, 1) dot (0, 1, 1, 2, 1, 4) = (0 + 1 + 1 + 0 + 0 + 4), where A051953 = (0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, ...) and (1, 1, 1, 0, 0, 1) = row 6 of A051731 where the 1's positions indicate the factors of 6. - Gary W. Adamson, Jul 11 2008
a(n) = A006128(n) - A220477(n) - n. - Omar E. Pol Jan 17 2013
a(n) = Sum_{i=1..floor(n/2)} i*(1-ceiling(frac(n/i))). - Wesley Ivan Hurt, Oct 25 2013
Dirichlet g.f.: zeta(s-1)*(zeta(s) - 1). - Ilya Gutkovskiy, Aug 07 2016
a(n) = 1 + A048050(n), n > 1. - R. J. Mathar, Mar 13 2018
Erdős (Elem. Math. 28 (1973), 83-86) shows that the density of even integers in the range of a(n) is strictly less than 1/2. The argument of Coppersmith (1987) shows that the range of a(n) has density at most 47/48 < 1. - N. J. A. Sloane, Dec 21 2019
G.f.: Sum_{k >= 2} x^k/(1 - x^k)^2. Cf. A296955. (This follows from the fact that if g(z) = Sum_{n >= 1} a(n)*z^n and f(z) = Sum_{n >= 1} a(n)*z^(N*n)/(1 - z^n) then f(z) = Sum_{k >= N} g(z^k), taking a(n) = n and N = 2.) - Peter Bala, Jan 13 2021
Faster converging g.f.: Sum_{n >= 1} q^(n*(n+1))*(n*q^(3*n+2) - (n + 1)*q^(2*n+1) - (n - 1)*q^(n+1) + n)/((1 - q^n)*(1 - q^(n+1))^2). (In equation 1 in Arndt, after combining the two n = 0 summands to get -t/(1 - t), apply the operator t*d/dt to the resulting equation and then set t = q and x = 1.) - Peter Bala, Jan 22 2021
a(n) = Sum_{d|n} d * (1 - [n = d]), where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021
a(n) = Sum_{i=1..n} ((n-1) mod i) - (n mod i). [See also A176079.] - José de Jesús Camacho Medina, Feb 23 2021

A000593 Sum of odd divisors of n.

Original entry on oeis.org

1, 1, 4, 1, 6, 4, 8, 1, 13, 6, 12, 4, 14, 8, 24, 1, 18, 13, 20, 6, 32, 12, 24, 4, 31, 14, 40, 8, 30, 24, 32, 1, 48, 18, 48, 13, 38, 20, 56, 6, 42, 32, 44, 12, 78, 24, 48, 4, 57, 31, 72, 14, 54, 40, 72, 8, 80, 30, 60, 24, 62, 32, 104, 1, 84, 48, 68, 18, 96, 48, 72, 13, 74, 38, 124

Offset: 1

N. J. A. Sloane

Denoted by Delta(n) or Delta_1(n) in Glaisher 1907. - Michael Somos, May 17 2013
A069289(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015
A000203, A001227 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
For the g.f.s given below by Somos Oct 29 2005, Jovovic, Oct 11 2002 and Arndt, Nov 09 2010, see the Hardy-Wright reference, proof of Theorem 382, p. 312, with x^2 replaced by x. - Wolfdieter Lang, Dec 11 2016
a(n) is also the total number of parts in all partitions of n into an odd number of equal parts. - Omar E. Pol, Jun 04 2017
It seems that a(n) divides A000203(n) for every n. - Ivan N. Ianakiev, Nov 25 2017 [Yes, see the formula dated Dec 14 2017].
Also, alternating row sums of A126988. - Omar E. Pol, Feb 11 2018
Where a(n) shows the number of equivalence classes of Hurwitz quaternions with norm n (equivalence defined by right multiplication with one of the 24 Hurwitz units as in A055672), A046897(n) seems to give the number of equivalence classes of Lipschitz quaternions with norm n (equivalence defined by right multiplication with one of the 8 Lipschitz units). - R. J. Mathar, Aug 03 2025

			G.f. = x + x^2 + 4*x^3 + x^4 + 6*x^5 + 4*x^6 + 8*x^7 + x^8 + 13*x^9 + 6*x^10 + 12*x^11 + ...

Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 496, pp. 69-246, Ellipses, Paris, 2004.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003, p. 312.
Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and Modular Forms, Vieweg, 1994, p. 133.
John Riordan, Combinatorial Identities, Wiley, 1968, p. 187.
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
Francesca Aicardi, Matricial formulas for partitions, arXiv:0806.1273 [math.NT], 2008.
Michael Baake and Robert V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999; Canad. J. Math. 51 (1999), 1258-1276.
John A. Ewell, On the sum-of-divisors function, Fib. Q., 45 (2007), 205-207.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
Kaya Lakein and Anne Larsen, A Proof of Merca's Conjectures on Sums of Odd Divisor Functions, arXiv:2107.07637 [math.NT], 2021.
Mircea Merca, The Lambert series factorization theorem, The Ramanujan Journal, January 2017, also here
Mircea Merca, Congruence identities involving sums of odd divisors function, Proceedings of the Romanian Academy, Series A, Volume 22, Number 2/2021, pp. 119-125.
Hossein Movasati and Younes Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv:1603.09411 [math.AG], 2016.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.
N. J. A. Sloane, Transforms.
H. J. Stephen Smith, Report on the Theory of Numbers. — Part VI., Report of the 35 Meeting of the British Association for the Advancement of Science (1866). See p. 336.
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Eric Weisstein's World of Mathematics, Partition Function Q.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
Index entries for "core" sequences.
Index entries for sequences mentioned by Glaisher.

Cf. A000005, A000203, A000265, A001227, A006128, A050999, A051000, A051001, A051002, A065442, A078471 (partial sums), A069289, A247837 (subset of the primes).

Cf. A301799, A301800.

a000593 = sum . a182469_row  -- Reinhard Zumkeller, May 01 2012, Jul 25 2011

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[j*x^j/(1+x^j): j in [1..2*m]])  )); // G. C. Greubel, Nov 07 2018

[&+[d:d in Divisors(n)|IsOdd(d)]:n in [1..75]]; // Marius A. Burtea, Aug 12 2019

A000593 := proc(n) local d,s; s := 0; for d from 1 by 2 to n do if n mod d = 0 then s := s+d; fi; od; RETURN(s); end;

Table[a := Select[Divisors[n], OddQ[ # ]&]; Sum[a[[i]], {i, 1, Length[a]}], {n, 1, 60}] (* Stefan Steinerberger, Apr 01 2006 *)
f[n_] := Plus @@ Select[ Divisors@ n, OddQ]; Array[f, 75] (* Robert G. Wilson v, Jun 19 2011 *)
a[ n_] := If[ n < 1, 0, Sum[ -(-1)^d n / d, {d, Divisors[ n]}]]; (* Michael Somos, May 17 2013 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# n / # &]]; (* Michael Somos, May 17 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] d, {d, Divisors[ n]}]]; (* Michael Somos, May 17 2013 *)
a[ n_] := If[ n < 1, 0, Times @@ (If[ # < 3, 1, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger @ n)]; (* Michael Somos, Aug 15 2015 *)
Array[Total[Divisors@ # /. d_ /; EvenQ@ d -> Nothing] &, {75}] (* Michael De Vlieger, Apr 07 2016 *)
Table[SeriesCoefficient[n Log[QPochhammer[-1, x]], {x, 0, n}], {n, 1, 75}] (* Vladimir Reshetnikov, Nov 21 2016 *)
Table[DivisorSum[n,#&,OddQ[#]&],{n,80}] (* Harvey P. Dale, Jun 19 2021 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(d+1) * n/d))}; /* Michael Somos, May 29 2005 */

N=66; x='x+O('x^N); Vec( serconvol( log(prod(j=1,N,1+x^j)), sum(j=1,N,j*x^j)))  /* Joerg Arndt, May 03 2008, edited by M. F. Hasler, Jun 19 2011 */

s=vector(100);for(n=1,100,s[n]=sumdiv(n,d,d*(d%2)));s /* Zak Seidov, Sep 24 2011*/

a(n)=sigma(n>>valuation(n,2)) \\ Charles R Greathouse IV, Sep 09 2014

from math import prod
from sympy import factorint
def A000593(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if p > 2) # Chai Wah Wu, Sep 09 2021

[sum(k for k in divisors(n) if k % 2) for n in (1..75)] # Giuseppe Coppoletta, Nov 02 2016

Inverse Moebius Transform of [0, 1, 0, 3, 0, 5, ...].
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s)).
a(2*n) = A000203(2*n)-2*A000203(n), a(2*n+1) = A000203(2*n+1). - Henry Bottomley, May 16 2000
a(2*n) = A054785(2*n) - A000203(2*n). - Reinhard Zumkeller, Apr 23 2008
Multiplicative with a(p^e) = 1 if p = 2, (p^(e+1)-1)/(p-1) if p > 2. - David W. Wilson, Aug 01 2001
a(n) = Sum_{d divides n} (-1)^(d+1)*n/d, Dirichlet convolution of A062157 with A000027. - Vladeta Jovovic, Sep 06 2002
Sum_{k=1..n} a(k) is asymptotic to c*n^2 where c=Pi^2/24. - Benoit Cloitre, Dec 29 2002
G.f.: Sum_{n>0} n*x^n/(1+x^n). - Vladeta Jovovic, Oct 11 2002
G.f.: (theta_3(q)^4 + theta_2(q)^4 -1)/24.
G.f.: Sum_{k>0} -(-x)^k / (1 - x^k)^2. - Michael Somos, Oct 29 2005
a(n) = A050449(n)+A050452(n); a(A000079(n))=1; a(A005408(n))=A000203(A005408(n)). - Reinhard Zumkeller, Apr 18 2006
From Joerg Arndt, Nov 09 2010: (Start)
G.f.: Sum_{n>=1} (2*n-1) * q^(2*n-1) / (1-q^(2*n-1)).
G.f.: deriv(log(P)) = deriv(P)/P where P = Product_{n>=1} (1 + q^n). (End)
Dirichlet convolution of A000203 with [1,-2,0,0,0,...]. - R. J. Mathar, Jun 28 2011
a(n) = Sum_{k = 1..A001227(n)} A182469(n,k). - Reinhard Zumkeller, May 01 2012
G.f.: -1/Q(0), where Q(k) = (x-1)*(1-x^(2*k+1)) + x*(-1 +x^(k+1))^4/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
a(n) = Sum_{k=1..n} k*A000009(k)*A081362(n-k). - Mircea Merca, Feb 26 2014
a(n) = A000203(n) - A146076(n). - Omar E. Pol, Apr 05 2016
a(2*n) = a(n). - Giuseppe Coppoletta, Nov 02 2016
a(n) = n * [x^n] log((-1; x)inf), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 21 2016
From Wolfdieter Lang, Dec 11 2016: (Start)
G.f.: Sum_{n>=1} x^n*(1+x^(2*n))/(1-x^(2*n))^2, from the second to last equation of the proof to Theorem 382 (with x^2 -> x) of the Hardy-Wright reference, p. 312.
a(n) = Sum_{d|n} (-d)*(-1)^(n/d), commutating factors of the D.g.f. given above by Jovovic, Oct 11 2002. See also the a(n) version given by Jovovic, Sep 06 2002. (End)
a(n) = A000203(n)/A038712(n). - Omar E. Pol, Dec 14 2017
a(n) = A000203(n)/(2^(1 + (A183063(n)/A001227(n))) - 1). - Omar E. Pol, Nov 06 2018
a(n) = A000203(2n) - 2*A000203(n). - Ridouane Oudra, Aug 28 2019
From Peter Bala, Jan 04 2021: (Start)
a(n) = (2/3)*A002131(n) + (1/3)*A002129(n) = (2/3)*A002131(n) + (-1)^(n+1)*(1/3)*A113184(n).
a(n) = A002131(n) - (1/2)*A146076; a(n) = 2*A002131(n) - A000203(n). (End)
a(n) = A000203(A000265(n)) - John Keith, Aug 30 2021
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000203(k) = A065442 - 1 = 0.60669... . - Amiram Eldar, Dec 14 2024

A001792 a(n) = (n+2)*2^(n-1).

Original entry on oeis.org

1, 3, 8, 20, 48, 112, 256, 576, 1280, 2816, 6144, 13312, 28672, 61440, 131072, 278528, 589824, 1245184, 2621440, 5505024, 11534336, 24117248, 50331648, 104857600, 218103808, 452984832, 939524096, 1946157056, 4026531840, 8321499136, 17179869184, 35433480192

Offset: 0

N. J. A. Sloane

Number of parts in all compositions (ordered partitions) of n + 1. For example, a(2) = 8 because in 3 = 2 + 1 = 1 + 2 = 1 + 1 + 1 we have 8 parts. Also number of compositions (ordered partitions) of 2n + 1 with exactly 1 odd part. For example, a(2) = 8 because the only compositions of 5 with exactly 1 odd part are 5 = 1 + 4 = 2 + 3 = 3 + 2 = 4 + 1 = 1 + 2 + 2 = 2 + 1 + 2 = 2 + 2 + 1. - Emeric Deutsch, May 10 2001
Binomial transform of natural numbers [1, 2, 3, 4, ...].
For n >= 1 a(n) is also the determinant of the n X n matrix with 3's on the diagonal and 1's elsewhere. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001
The arithmetic mean of first n terms of the sequence is 2^(n-1). - Amarnath Murthy, Dec 25 2001, corrected by M. F. Hasler, Dec 17 2016
Also the number of "winning paths" of length n across an n X n Hex board. Satisfies the recursion a(n) = 2a(n-1) + 2^(n-2). - David Molnar (molnar(AT)stolaf.edu), Apr 10 2002
Diagonal in A053218. - Benoit Cloitre, May 08 2002
Let M_n be the n X n matrix m_(i, j) = 1 + abs(i-j) then det(M_n) = (-1)^(n-1)*a(n-1). - Benoit Cloitre, May 28 2002
Absolute value of determinant of n X n matrix of form: [1 2 3 4 5 / 2 1 2 3 4 / 3 2 1 2 3 / 4 3 2 1 2 / 5 4 3 2 1]. - Benoit Cloitre, Jun 20 2002
Number of ones in all (n+1)-bit integers (cf. A000120). - Ralf Stephan, Aug 02 2003
This sequence also emerges as a floretion force transform of powers of 2 (see program code). Define a(-1) = 0 (as the sequence is returned by FAMP). Then a(n-1) + A098156(n+1) = 2*a(n) (conjecture). - Creighton Dement, Mar 14 2005
This sequence gives the absolute value of the determinant of the Toeplitz matrix with first row containing the first n integers. - Paul Max Payton, May 23 2006
Equals sums of rows right of left edge of A102363 divided by three, + 2^K. - David G. Williams (davidwilliams(AT)paxway.com), Oct 08 2007
If X_1, X_2, ..., X_n are 2-blocks of a (2n+1)-set X then, for n >= 1, a(n) is the number of (n+1)-subsets of X intersecting each X_i, (i = 1, 2, ..., n). - Milan Janjic, Nov 18 2007
Also, a(n-1) is the determinant of the n X n matrix with A[i, j] = n - |i-j|. - M. F. Hasler, Dec 17 2008
1/2 the number of permutations of 1 .. (n+2) arranged in a circle with exactly one local maximum. - R. H. Hardin, Apr 19 2009
The first corrector line for transforming 2^n offset 0 with a leading 1 into the Fibonacci sequence. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009
a(n) is the number of runs of consecutive 1's in all binary sequences of length (n+1). - Geoffrey Critzer, Jul 02 2009
Let X_j (0 < j <= 2^n) all the subsets of N_n; m(i, j) := if {i} in X_j then 1 else 0. Let A = transpose(M).M; Then a(i, j) = (number of elements)|X_i intersect X_j|. Determinant(X*I-A) = (X-(n+1)*2^(n-2))*(X-2^(n-2))^(n-1)*X^(2^n-n).
  Eigenvector for (n+1)*2^(n-2) is V_i=|X_i|.
  Sum_{k=1..2^n} |X_i intersect X_k|*|X_k| = (n+1)*2^(n-2)*|X_i|.
  Eigenvectors for 2^(n-2) are {line(M)[i] - line(M)[j], 1 <= i, j <= n}. - CLARISSE Philippe (clarissephilippe(AT)yahoo.fr), Mar 24 2010
The sequence b(n) = 2*A001792(n), for n >= 1 with b(0) = 1, is an elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 187, 190, 250 and 442, lead to the b(n) sequence. For the corner squares these vectors lead to the companion sequence A134401. - Johannes W. Meijer, Aug 15 2010
Equals partial sums of A045623: (1, 2, 5, 12, 28, ...); where A045623 = the convolution square of (1, 1, 2, 4, 8, 16, 32, ...). - Gary W. Adamson, Oct 26 2010
Equals (1, 2, 4, 8, 16, ...) convolved with (1, 1, 2, 4, 8, 16, ...); e.g., a(3) = 20 = (1, 1, 2, 4) dot (8, 4, 2, 1) = (8 + 4 + 4 + 4). - Gary W. Adamson, Oct 26 2010
This sequence seems to give the first x+1 nonzero terms in the sequence derived by subtracting the m-th term in the x_binacci sequence (where the first term is one and the y-th term is the sum of x terms immediately preceding it) from 2^(m-2). - Dylan Hamilton, Nov 03 2010
Recursive formulas for a(n) are in many cases derivable from its property wherein delta^k(a(n)) - a(n) = k*2^n where delta^k(a(n)) represents the k-th forward difference of a(n). Provable with a difference table and a little induction. - Ethan Beihl, May 02 2011
Let f(n,k) be the sum of numbers in the subsets of size k of {1, 2, ..., n}. Then a(n-1) is the average of the numbers f(n, 0), ... f(n, n). Example: (f(3, 1), f(3, 2), f(3, 3)) = (6, 12, 6), with average (6+12+6)/3. - Clark Kimberling, Feb 24 2012
a(n) is the number of length-2n binary sequences that contain a subsequence of ones with length n or more. To derive this result, note that there are 2^n sequences where the initial one of the subsequence occurs at entry one. If the initial one of the subsequence occurs at entry 2, 3, ..., or n + 1, there are 2^(n-1) sequences since a zero must precede the initial one. Hence a(n) = 2^n + n*2^(n-1)=(n+2)2^(n-1). An example is given in the example section below. - Dennis P. Walsh, Oct 25 2012
As the total number of parts in all compositions of n+1 (see the first line in Comments) the equivalent sequence for partitions is A006128. On the other hand, as the first differences of A001787 (see the first line in Crossrefs) the equivalent sequence for partitions is A138879. - Omar E. Pol, Aug 28 2013
a(n) is the number of spanning trees of the complete tripartite graph K_{n,1,1}. - James Mahoney, Oct 24 2013
a(n-1) = denominator of the mean (2n/(n+1), after reduction), of the compositions of n; numerator is given by A022998(n). - Clark Kimberling, Mar 11 2014
From Tom Copeland, Nov 09 2014: (Start)
The shifted array belongs to an interpolated family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the interpolating o.g.f. (1-sqrt(1-4x/(1+(1-t)x)))/2 and inverse x(1-x)/(1+(t-1)x(1-x)). See A091867 for more info on this family. Here the interpolation is t=-3 (mod signs in the results).
Let C(x) = (1 - sqrt(1-4x))/2, an o.g.f. for the Catalan numbers A000108, with inverse Cinv(x) = x*(1-x) and P(x,t) = x/(1+t*x) with inverse P(x,-t).
Shifted o.g.f: G(x) = x*(1-x)/(1 - 4x*(1-x)) = P[Cinv(x),-4].
Inverse o.g.f: Ginv(x) = [1 - sqrt(1 - 4*x/(1+4x))]/2 = C[P(x, 4)] (signed shifted A001700). Cf. A030528. (End)
For n > 0, element a(n) of the sequence is equal to the gradients of the (n-1)-th row of Pascal triangle multiplied with the square of the integers from n+1,...,1. I.e., row 3 of Pascal's triangle 1,3,3,1 has gradients 1,2,0,-2,-1, so a(4) = 1*(5^2) + 2*(4^2) + 0*(3^2) - 2*(2^2) - 1*(1^2) = 48. - Jens Martin Carlsson, May 18 2017
Number of self-avoiding paths connecting all the vertices of a convex (n+2)-gon. - Ivaylo Kortezov, Jan 19 2020
a(n-1) is the total number of elements of subsets of {1,2,..,n} that contain n.  For example, for n = 3, a(2) = 8, and the subsets of {1,2,3} that contain 3 are {3}, {1,3}, {2,3}, {1,2,3}, with a total of 8 elements. - Enrique Navarrete, Aug 01 2020

			a(0) = 1, a(1) = 2*1 + 1 = 3, a(2) = 2*3 + 2 = 8, a(3) = 2*8 + 4 = 20, a(4) = 2*20 + 8 = 48, a(5) = 2*48 + 16 = 112, a(6) = 2*112 + 32 = 256, ... - _Philippe Deléham_, Apr 19 2009
a(2) = 8 since there are 8 length-4 binary sequences with a subsequence of ones of length 2 or more, namely, 1111, 1110, 1101, 1011, 0111, 1100, 0110, and 0011. - _Dennis P. Walsh_, Oct 25 2012
G.f. = 1 + 3*x + 8*x^2 + 20*x^3 + 48*x^4 + 112*x^5 + 256*x^6 + 576*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
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).
A. M. Stepin and A. T. Tagi-Zade, Words with restrictions, pp. 67-74 of Kvant Selecta: Combinatorics I, Amer. Math. Soc., 2001 (G_n on p. 70).

T. D. Noe, Table of n, a(n) for n = 0..500
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].
Marco Abrate, Stefano Barbero, Umberto Cerruti and Nadir Murru, Colored compositions, Invert operator and elegant compositions with the "black tie", Discrete Math., Vol. 335 (2014), pp. 1-7. MR3248794.
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Colored compositions, Invert operator and elegant compositions with the "black tie", arXiv:1409.6454 [math.NT], 2014.
Milica Andelic, C. M. da Fonseca and A. Pereira, The mu-permanent, a new graph labeling, and a known integer sequence, arXiv preprint arXiv:1609.04208 [math.CO], 2016.
Félix Balado and Guénolé C. M. Silvestre, Runs of Ones in Binary Strings, arXiv:2302.11532 [math.CO], 2023. See pp. 6-7.
Jean-Luc Baril and Nathanaël Hassler, Intervals in a family of Fibonacci lattices, Univ. de Bourgogne (France, 2024). See p. 7.
Neil J. Calkin, A curious binomial identity, Discr. Math., Vol. 131, No. 1-3 (1994), pp. 335-337.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs., Vol. 3 (2000), #00.1.5.
Filippo Disanto and Simone Rinaldi, Symmetric convex permutominoes and involutions, PU. M. A., Vol. 22, No. 1 (2011), pp. 39-60.
Frank Ellermann, Illustration of binomial transforms.
David Eppstein, Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time, arXiv:2303.00147 [cs.CG], 2023, pp. 2, 19.
Guillermo Esteban, Clemens Huemer and Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.
M. Hirschhorn, Calkin's binomial identity, Discr. Math., Vol. 159, No. 1-3 (1996), pp. 273-278.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 146.
Milan Janjić, Two Enumerative Functions
Milan Janjić and Boris Petković, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
Milan Janjić and Boris Petković, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
C. W. Jones, J. C. P. Miller, J. F. C. Conn and R. C. Pankhurst, Tables of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
Sergey Kitaev and Jeffrey B. Remmel, A note on p-Ascent Sequences, Preprint, 2016.
Sergey Kitaev and Jeffrey Remmel, p-Ascent Sequences, arXiv preprint arXiv:1503.00914 [math.CO], 2015.
Ivaylo Kortezov, problem 8.4 ("Задача 8.4" in Bulgarian) in National Math Contest "Atanas Radev" 2020.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Maohua Le, Two Classes of Smarandache Determinants, Scientia Magna, Vol. 2, No. 1 (2006), pp. 20-25.
Donatella Merlini and Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021.
John Riordan and N. J. A. Sloane, Correspondence, 1974.
N. J. A. Sloane, Transforms.
I. Tasoulas, K. Manes, and A. Sapounakis, Hamiltonian intervals in the lattice of binary paths, Elect. J. Comb. (2024) Vol. 31, Issue 1, P1.39.
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., Vol. 274 (2004), 331-342.
Index entries for linear recurrences with constant coefficients, signature (4,-4).
Index entries for sequences related to Chebyshev polynomials.

First differences of A001787.
a(n) = A049600(n, 1), a(n) = A030523(n + 1, 1).
Cf. A053113.
Row sums of triangles A008949 and A055248.
a(n) = -A039991(n+2, 2).
Cf. A130584, A125092, A128064, A000079, A164910, A045623, A000120, A053120.
Cf. A000108, A005043, A091867, A001700, A030528.
If the exponent E in a(n) = Sum_{m=0..n} (Sum_{k=0..m} C(n,k))^E is 1, 2, 3, 4, 5 we get A001792, A003583, A007403, A294435, A294436 respectively.

List([0..35],n->(n+2)*2^(n-1)); # Muniru A Asiru, Sep 25 2018

a001792 n = a001792_list !! n
a001792_list = scanl1 (+) a045623_list
-- Reinhard Zumkeller, Jul 21 2013

[(n+2)*2^(n-1): n in [0..40]]; // Vincenzo Librandi, Nov 10 2014

A001792 := n-> (n+2)*2^(n-1);
spec := [S, {B=Set(Z, 0 <= card), S=Prod(Z, B, B)}, labeled]: seq(combstruct[count](spec, size=n)/4, n=2..30); # Zerinvary Lajos, Oct 09 2006
A001792:=-(-3+4*z)/(2*z-1)^2; # Simon Plouffe in his 1992 dissertation, which gives the sequence without the initial 1
G(x):=1/exp(2*x)*(1-x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(abs(f[n]),n=0..28 ); # Zerinvary Lajos, Apr 17 2009
a := n -> hypergeom([-n, 2], [1], -1);
seq(round(evalf(a(n),32)), n=0..31); # Peter Luschny, Aug 02 2014

matrix[n_Integer /; n >= 1] := Table[Abs[p - q] + 1, {q, n}, {p, n}]; a[n_Integer /; n >= 1] := Abs[Det[matrix[n]]] (* Josh Locker (joshlocker(AT)macfora.com), Apr 29 2004 *)
g[n_,m_,r_] := Binomial[n - 1, r - 1] Binomial[m + 1, r] r; Table[1 + Sum[g[n, k - n, r], {r, 1, k}, {n, 1, k - 1}], {k, 1, 29}] (* Geoffrey Critzer, Jul 02 2009 *)
a[n_] := (n + 2)*2^(n - 1); a[Range[0, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{4, -4}, {1, 3}, 40] (* Harvey P. Dale, Aug 29 2011 *)
CoefficientList[Series[(1 - x) / (1 - 2 x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)
b[i_]:=i; a[n_]:=Abs[Det[ToeplitzMatrix[Array[b, n], Array[b, n]]]]; Array[a, 40] (* Stefano Spezia, Sep 25 2018 *)
a[n_]:=Hypergeometric2F1[2,-n+1,1,-1];Array[a,32] (* Giorgos Kalogeropoulos, Jan 04 2022 *)

A001792(n)=(n+2)<<(n-1) \\ M. F. Hasler, Dec 17 2008

for n in range(0,40): print(int((n+2)*2**(n-1)), end=' ') # Stefano Spezia, Oct 16 2018

a(n) = (n+2)*2^(n-1).
G.f.: (1 - x)/(1 - 2*x)^2 = 2F1(1,3;2;2x).
a(n) = 4*a(n-1) - 4*a(n-2).
G.f. (-1 + (1-2*x)^(-2))/(x*2^2). - Wolfdieter Lang
a(n) = A018804(2^n). - Matthew Vandermast, Mar 01 2003
a(n) = Sum_{k=0..n+2} binomial(n+2, 2k)*k. - Paul Barry, Mar 06 2003
a(n) = (1/4)*A001787(n+2). - Emeric Deutsch, May 24 2003
With a leading 0, this is ((n+1)2^n - 0^n)/4 = Sum_{m=0..n} binomial(n - 1, m - 1)*m, the binomial transform of A004526(n+1). - Paul Barry, Jun 05 2003
a(n) = Sum_{k=0..n} binomial(n, k)*(k + 1). - Lekraj Beedassy, Jun 24 2004
a(n) = A000244(n) - A066810(n). - Ross La Haye, Apr 29 2006
Row sums of triangle A130585. - Gary W. Adamson, Jun 06 2007
Equals A125092 * [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, Nov 16 2007
a(n) = (n+1)*2^n - n*2^(n-1). Equals A128064 * A000079. - Gary W. Adamson, Dec 28 2007
G.f.: F(3, 1; 2; 2x). - Paul Barry, Sep 03 2008
a(n) = 1 + Sum_{k=1..n} (n - k + 4)2^(n - k - 1). This follows from the result that the number of parts equal to k in all compositions of n is (n - k + 3)2^(n - k - 2) for 0 < k < n. - Geoffrey Critzer, Sep 21 2008
a(n) = 2^(n-1) + 2 a(n-1) ; a(n-1) = det(n - |i - j|){i, j = 1..n}. - _M. F. Hasler, Dec 17 2008
a(n) = 2*a(n-1) + 2^(n-1). - Philippe Deléham, Apr 19 2009
a(n) = A164910(2^n). - Gary W. Adamson, Aug 30 2009
a(n) = Sum_{i=1..2^n} gcd(i, 2^n) = A018804(2^n). So we have: 2^0 * phi(2^n) + ... + 2^n * phi(2^0) = (n + 2)*2^(n-1), where phi is the Euler totient function. - Jeffrey R. Goodwin, Nov 11 2011
a(n) = Sum_{j=0..n} Sum_{i=0..n} binomial(n, i + j). - Yalcin Aktar, Jan 17 2012
Eigensequence of an infinite lower triangular matrix with 2^n as the left border and the rest 1's. - Gary W. Adamson, Jan 30 2012
G.f.: 1 + 2*x*U(0) where U(k) =  1 + (k + 1)/(2 - 8*x/(4*x + (k + 1)/U(k + 1))); (continued fraction, 3 - step). - Sergei N. Gladkovskii, Oct 19 2012
a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n,j). - Peter Luschny, Dec 03 2013
a(n) = Hyper2F1([-n, 2], [1], -1). - Peter Luschny, Aug 02 2014
G.f.: 1 / (1 - 3*x / (1 + x / (3 - 4*x))). - Michael Somos, Aug 26 2015
a(n) = -A053120(2+n, n), n >= 0, the negative of the third (sub)diagonal of the triangle of Chebyshev's T polynomials. - Wolfdieter Lang, Nov 26 2019
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=0} 1/a(n) = 8*log(2) - 4.
Sum_{n>=0} (-1)^n/a(n) = 4 - 8*log(3/2). (End)
E.g.f.: exp(2*x)*(1 + x). - Stefano Spezia, Jun 11 2021

A066186 Sum of all parts of all partitions of n.

Original entry on oeis.org

0, 1, 4, 9, 20, 35, 66, 105, 176, 270, 420, 616, 924, 1313, 1890, 2640, 3696, 5049, 6930, 9310, 12540, 16632, 22044, 28865, 37800, 48950, 63336, 81270, 104104, 132385, 168120, 212102, 267168, 334719, 418540, 520905, 647172, 800569, 988570, 1216215, 1493520

Offset: 0

Wouter Meeussen, Dec 15 2001

Sum of the zeroth moments of all partitions of n.
Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that any part z is composed of labeled elements of amount 1, i.e., z = 1_1 + 1_2 + ... + 1_z. Then one can take from z a single element in z different ways. E.g., for n=3 to n=2 we have A066186(3) = 9 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [111], [12] --> [111], [12] --> [2], [3] --> 2, [3] --> 2, [3] --> 2. For the unlabeled case, one can take a single element from z in only one way. Then the number of one-element transitions from the integer partitions of n to the partitions of n-1 is given by A000070. E.g., A000070(3) = 4 and for the transition from n=3 to n=2 one has [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. - Thomas Wieder, May 20 2004
Also sum of all parts of all regions of n (Cf. A206437). - Omar E. Pol, Jan 13 2013
From Omar E. Pol, Jan 19 2021: (Start)
Apart from initial zero this is also as follows:
Convolution of A000203 and A000041.
Convolution of A024916 and A002865.
For n >= 1, a(n) is also the number of cells in a symmetric polycube in which the terraces are the symmetric representation of sigma(k), for k = n..1, (cf. A237593) starting from the base and located at the levels A000041(0)..A000041(n-1) respectively. The polycube looks like a symmetric tower (cf. A221529). A dissection is a three-dimensional spiral whose top view is described in A239660. The growth of the volume of the polycube represents each convolution mentioned above. (End)
From Omar E. Pol, Feb 04 2021: (Start)
a(n) is also the sum of all divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned divisors are also all parts of all partitions of n.
Apart from initial zero this is also the convolution of A340793 and A000070. (End)

			a(3)=9 because the partitions of 3 are: 3, 2+1 and 1+1+1; and (3) + (2+1) + (1+1+1) = 9.
a(4)=20 because A000041(4)=5 and 4*5=20.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265, alternate copy. - From _N. J. A. Sloane_, Jan 02 2013
F. G. Garvan, Higher-order spt functions, arXiv:1008.1207 [math.NT], 2010.
T. J. Osler, A. Hassen and T. R. Chandrupatia, Surprising connections between partitions and divisors, The College Mathematics Journal, Vol. 38. No. 4, Sep. 2007, 278-287 (see p. 287).
Omar E. Pol, Illustration of a(10), prism and tower, each polycube contains 420 cubes.
Omar E. Pol, Illustration of initial terms of A066186 and of A139582 (n>=1)

Cf. A000041, A093694, A000070, A132825, A001787 (same for ordered partitions), A277029, A000203, A221529, A237593, A239660.
First differences give A138879. - Omar E. Pol, Aug 16 2013
Row sums of triangles A138785, A181187, A245099, A337209, A339106, A340423, A340424, A221529, A302246, A338156, A340035, A340056, A340057, A346741. - Omar E. Pol, Aug 02 2021

a066186 = sum . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

with(combinat): a:= n-> n*numbpart(n): seq(a(n), n=0..50); # Zerinvary Lajos, Apr 25 2007

PartitionsP[ Range[0, 60] ] * Range[0, 60]

a(n)=numbpart(n)*n \\ Charles R Greathouse IV, Mar 10 2012

from sympy import npartitions
def A066186(n): return n*npartitions(n) # Chai Wah Wu, Oct 22 2023

[n*Partitions(n).cardinality() for n in range(41)] # Peter Luschny, Jul 29 2014

a(n) = n * A000041(n). - Omar E. Pol, Oct 10 2011
G.f.: x * (d/dx) Product_{k>=1} 1/(1-x^k), i.e., derivative of g.f. for A000041. - Jon Perry, Mar 17 2004 (adjusted to match the offset by Geoffrey Critzer, Nov 29 2014)
Equals A132825 * [1, 2, 3, ...]. - Gary W. Adamson, Sep 02 2007
a(n) = A066967(n) + A066966(n). - Omar E. Pol, Mar 10 2012
a(n) = A207381(n) + A207382(n). - Omar E. Pol, Mar 13 2012
a(n) = A006128(n) + A196087(n). - Omar E. Pol, Apr 22 2012
a(n) = A220909(n)/2. - Omar E. Pol, Jan 13 2013
a(n) = Sum_{k=1..n} A000203(k)*A000041(n-k), n >= 1. - Omar E. Pol, Jan 20 2013
a(n) = Sum_{k=1..n} k*A036043(n,n-k+1). - L. Edson Jeffery, Aug 03 2013
a(n) = Sum_{k=1..n} A024916(k)*A002865(n-k), n >= 1. - Omar E. Pol, Jul 13 2014
a(n) ~ exp(Pi*sqrt(2*n/3))/(4*sqrt(3)) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Oct 24 2016
a(n) = Sum_{k=1..n} A340793(k)*A000070(n-k), n >= 1. - Omar E. Pol, Feb 04 2021

a(0) added by Franklin T. Adams-Watters, Jul 28 2014

A206437 Triangle read by rows: T(j,k) is the k-th part of the j-th region of the set of partitions of n, if 1 <= j <= A000041(n).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 2, 4, 2, 1, 1, 1, 3, 5, 2, 1, 1, 1, 1, 1, 2, 4, 2, 3, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 3, 5, 2, 4, 7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Feb 14 2012

Here the j-th "region" of the set of partitions of n (or more simply the j-th "region" of n) is defined to be the first h elements of the sequence formed by the smallest parts in nonincreasing order of the partitions of the largest part of the j-th partition of n, with the list of partitions in colexicographic order, where h = j - i, and i is the index of the previous partition of n whose largest part is greater than the largest part of the j-th partition of n, or i = 0 if such previous largest part does not exist. The largest part of the j-th region of n is A141285(j) and the number of parts is h = A194446(j).
Some properties of the regions of n:
- The number of regions of n equals the number of partitions of n (see A000041).
- The set of regions of n contain the sets of regions of all positive integers previous to n.
- The first j regions of n are also first j regions of all integers greater than n.
- The sums of all largest parts of all regions of n equals the total number of parts of all regions of n. See A006128(n).
- If T(j,1) is a record in the sequence then the leading diagonals of triangle formed by the first j rows give the partitions of n (see example).
- The rank of a region is the largest part minus the number of parts (see A194447).
- The sum of all ranks of the regions of n is equal to zero.
How to make a diagram of the regions and partitions of n: in the first quadrant of the square grid we draw a horizontal line {[0, 0],[n, 0]} of length n. Then we draw a vertical line {[0, 0],[0, p(n)]} of length p(n) where p(n) is the number of partitions of n. Then, for j = 1..p(n), we draw a horizontal line {[0, j],[g, j]} where g = A141285(j) is the largest part of the j-th partition of n, with the list of partitions in colexicographic order. Then, for n = 1 .. p(n), we draw a vertical line from the point [g,j] down to intercept the next segment in a lower row. So we have a number of closed regions. Then we divide each region of n in horizontal rectangles with shorter sides = 1. We can see that in the original rectangle of area n*p(n) each row contains a set of rectangles whose areas are equal to the parts of one of the partitions of n. Then each region of n is labeled according to the position of its largest part on axis "y". Note that each region of n is similar to a mirror version of the Young diagram of one of the partitions of s, where s is the sum of all parts of the region. See the illustrations of the seven regions of 5 in the Links section.
Note that if row j of triangle contains parts of size 1 then the parts of row j are the smallest parts of all partitions of T(j,1), (see A046746), and also T(j,1) is a record in the sequence and also j is the number of partitions of T(j,1), (see A000041). Otherwise, if row j does not contain parts of size 1 then the parts of row j are the emergent parts of the next record in the sequence (see A183152). Row j is also the partition of A186412(j).
Also triangle read by rows in which row r lists the parts of the last section of the set of partitions of r, ordered by regions, such that the previous parts to the part of size r are the emergent parts of the partitions of r (see A138152) and the rest are the smallest parts of the partitions of r (see example). - Omar E. Pol, Apr 28 2012

			-------------------------------------------
  Region j   Triangle of parts
-------------------------------------------
  1          1;
  2          2,1;
  3          3,1,1;
  4          2;
  5          4,2,1,1,1;
  6          3;
  7          5,2,1,1,1,1,1;
  8          2;
  9          4,2;
  10         3;
  11         6,3,2,2,1,1,1,1,1,1,1;
  12         3;
  13         5,2;
  14         4;
  15         7,3,2,2,1,1,1,1,1,1,1,1,1,1,1;
.
The rotated triangle shows each row as a partition:
                             7
                           4   3
                         5       2
                       3   2       2
                     6               1
                   3   3               1
                 4       2               1
               2   2       2               1
             5               1               1
           3   2               1               1
         4       1               1               1
       2   2       1               1               1
     3       1       1               1               1
   2   1       1       1               1               1
 1   1   1       1       1               1               1
.
Alternative interpretation of this sequence:
Triangle read by rows in which row r lists the parts of the last section of the set of partitions of r ordered by regions (see comments):
   [1];
   [2,1];
   [3,1,1];
   [2],[4,2,1,1,1];
   [3],[5,2,1,1,1,1,1];
   [2],[4,2],[3],[6,3,2,2,1,1,1,1,1,1,1];
   [3],[5,2],[4],[7,3,2,2,1,1,1,1,1,1,1,1,1,1,1];

Robert Price, Table of n, a(n) for n = 1..321, first 75 regions.
Omar E. Pol, Illustration of the seven regions of 5
Omar E. Pol, Illustration of initial terms, regions = 1..77 (2D view)
Omar E. Pol, Illustration of initial terms, regions = 1..30 (3D view)
Omar E. Pol, Visualization of regions in a diagram for A006128
Robert Price, Mathematica program to draw diagram up to n=28

Positive integers in A193870. Column 1 is A141285. Row j has length A194446(j). Row sums give A186412. Records are A000027.

Cf. A000041, A046746, A135010, A138121, A182699, A182703, A182709, A183152, A186114, A187219, A194436-A194439, A194447, A194448, A196025, A198381.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0, 2];
reg = {}; l = {};
For[j = 1, j <= 22, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[reg, Take[Reverse[First /@ lex[mx]], j - i]];
  ];
Flatten@reg  (* Robert Price, Apr 21 2020, revised Jul 24 2020 *)

Further edited by Omar E. Pol, Mar 31 2012, Jan 27 2013
Minor edits by Omar E. Pol, Apr 23 2020
Comments corrected (following a suggestion from Peter Munn) by Omar E. Pol, Jul 20 2025

cp's OEIS Frontend

A138137 First differences of A006128.

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A124920 Location of record values in A080577; also partial sums of A006128 plus 1.

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A285220 Indices of primes in A006128.

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A161981 A006128(n) mod n.

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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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A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

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