A013959 -id:A013959 - 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

A008408 Theta series of Leech lattice.

Original entry on oeis.org

1, 0, 196560, 16773120, 398034000, 4629381120, 34417656000, 187489935360, 814879774800, 2975551488000, 9486551299680, 27052945920000, 70486236999360, 169931095326720, 384163586352000, 820166620815360, 1668890090322000

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Third Edition, Springer-Verlag,1993, pp. 51, 134-135.
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 113.
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 501 terms from N. J. A. Sloane)
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, The sphere packing problem in dimension 24, arXiv:1603.06518 [math.NT], 2016.
Henry Cohn and Stephen D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24, arXiv:1603.04759 [math.MG], 2016
David de Laat and Frank Vallentin, A Breakthrough in Sphere Packing: The Search for Magic Functions, arXiv preprint arXiv:1607.02111 [math.MG], 2016.
Fern Gossow, Lyndon-like cyclic sieving and Gauss congruence, arXiv:2410.05678 [math.CO], 2024. See p. 26.
Nadia Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
G. Nebe and N. J. A. Sloane, Home page for lattice
K. Ono, S. Robins and P. T. Wahl, On the Representation of Integers as Sums of Triangular Numbers, (see p. 12), Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Seven Staggering Sequences.
Eric Weisstein's World of Mathematics, Leech Lattice.
Eric Weisstein's World of Mathematics, Theta Series.

Cf. A004009, A108093, A000594, A108093 (24th root), A034597, A034598, A198343, A013959.

// Theta series of the Leech lattice, from John Cannon, Dec 29 2006
A008408Q := function(prec) M12 := ModularForms(Gamma0(1), 12); t1 := Basis(M12)[1]; T := PowerSeries(t1, prec); return Coefficients(T); end function; Q := A008408Q(1000); Q[678];

Basis( ModularForms( Gamma0(1), 12), 30) [1] ; /* Michael Somos, Jun 09 2014 */

with(numtheory); f := 1+240*add(sigma[ 3 ](m)*q^(2*m),m=1..50); t := q^2*mul((1-q^(2*m))^24,m=1..50); series(f^3-720*t,q,51);

max = 17; f = 1 + 240*Sum[ DivisorSigma[3, m]*q^(2m), {m, 1, max}]; t = q^2*Product[(1 - q^(2m))^24, {m, 1, max}]; Partition[ CoefficientList[ Series[f^3 - 720t, {q, 0, 2 max}], q], 2][[All, 1]] (* Jean-François Alcover , Oct 14 2011, after Maple *)
(* From version 6 on *) f[q_] = LatticeData["Leech", "ThetaSeriesFunction"][x] /. x -> -I*Log[q]/Pi; Series[f[q], {q, 0, 32}] // CoefficientList[#, q^2]& (* Jean-François Alcover, May 15 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], SeriesCoefficient[(1 + 240 Sum[ q^k DivisorSigma[ 3, k], {k, n}])^3 - 720 q QPochhammer[ q]^24, {q, 0, n}]]; (* Michael Somos, Jun 09 2014 *)

{a(n) = if( n<1, n==0, polcoeff( 1 + (sum(k=1, n, sigma(k,11)*x^k) - x*eta(x + O(x^n))^24) * 65520/691, n))}; /* Michael Somos, Oct 19 2006 */

{a(n) = if( n<1, n==0, polcoeff( sum(k=1, n, 240*sigma(k,3)*x^k, 1 + x*O(x^n))^3 - 720*x*eta(x + O(x^n))^24, n))}; /* Michael Somos, Oct 19 2006 */

from sympy import divisor_sigma
def A008408(n): return 65520*(divisor_sigma(n,11)-(n**4*divisor_sigma(n)-24*((m:=n+1>>1)**2*(0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*divisor_sigma(m)**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2) - 20*n**3) + n**4)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m)))))//691 if n else 1 # Chai Wah Wu, Nov 17 2022

A = ModularForms( Gamma0(1), 12, prec=30) . basis() ; A[1] - 65520/691*A[0] # Michael Somos, Jun 09 2014

The simplest way to obtain this is to take the cube of the theta series for E_8 (A004009) and subtract 720 times the g.f. for the Ramanujan numbers (A000594).
This theta series is thus also the q-expansion of (7/12) E_4(z)^3 + (5/12) E_6(z)^2. Cf. A013973. - Daniel D. Briggs, Nov 25 2011
a(n) = 65520*(A013959(n) - A000594(n))/691, n >= 1. a(0) = 1. Expansion of the Theta series of the Leech lattice in powers of q^2. See the Conway and Sloane reference. - Wolfdieter Lang, Jan 16 2017

A046694 Ramanujan tau numbers mod 691 = sum of 11th power of divisors mod 691.

Original entry on oeis.org

1, 667, 252, 601, 684, 171, 531, 178, 372, 168, 469, 123, 629, 385, 309, 388, 611, 55, 672, 630, 449, 491, 92, 632, 57, 106, 88, 580, 173, 185, 366, 666, 27, 538, 429, 379, 622, 456, 269, 136, 87, 280, 36, 632, 160, 556, 435, 345, 194, 14, 570, 52, 209, 652, 172, 542, 49

Offset: 1

Marc LeBrun

Ramanujan tau is multiplicative, so this sequence is multiplicative mod 691.
There are pairs of identical terms a(n) and a(n+1). The first such twin pair is a(184) = a(185) = 483. The indices for a first twin in a pair are listed in A121733. Corresponding twin values are listed in A121734. - Alexander Adamchuk, Aug 18 2006
Set of values of a(n) consists of all integers from 0 to 690. The first a(n) = 0 occur at n = 2*691 - 1 = 1381 that is a prime. Set of numbers n such that a(n) = 0 is a union of all terms of the arithmetic progressions k*p, where p is a prime of the form p = 2m*691 - 1 and k>0 is an integer. Primes of the form p = 2m*691 - 1 are listed in A134671 = {1381,5527,8291,12437,22111,29021,30403,...}. It appears that in a(n) there are strings of consecutive zeros of any length. The first pair of consecutive zeros occurs at n = {16581,16582}. The least numbers k such that a(n) has a string of n consecutive zeros starting with a(k) are listed in A134670(n) = {1381,16581,290217,1409635,...}. - Alexander Adamchuk, Nov 05 2007

			Coefficient of x^2 in tau(x) = -24; 1^11+2^11 = 2049 = 667 mod 691 = -24 mod 691.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 169, (10.6.4).

T. D. Noe, Table of n, a(n) for n = 1..10000
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.

Cf. A000594, A013959, A121733, A121734, A098108, A126812-...

Cf. also A134670, A134671, A121742, A121743, A262339.

A046694 := proc(n)
    numtheory[sigma][11](n) mod 691 ;
end proc: # R. J. Mathar, Feb 01 2013

a[n_] := Mod[Total[Divisors[n]^11], 691]; a /@ Range[57] (* Jean-François Alcover , Apr 22 2011 *)
Table[Mod[DivisorSigma[11,n],691],{n,60}] (* Harvey P. Dale, Jun 01 2012 *)

a(n)=ramanujantau(n)%691 \\ Charles R Greathouse IV, Feb 08 2017

a(n)=sigma(n,11)%691 \\ Charles R Greathouse IV, Sep 09 2022

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 11)%691 # Indranil Ghosh, Apr 24 2017

a(n) = A000594(n) mod A262339(6). - Jonathan Sondow, Sep 22 2015

A015769 Numbers k such that phi(k) | sigma_11(k).

Original entry on oeis.org

1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190, 210, 235, 248, 264, 270, 329, 357, 418, 420, 470, 570, 594, 616, 630, 658, 695, 705, 714, 744, 799, 807, 812, 840, 910, 940, 987, 1045, 1240, 1254, 1316, 1390, 1410, 1485, 1529

Offset: 1

Robert G. Wilson v

nonn

sigma_11(n) = A013959(n) is the sum of the 11th powers of the divisors of n.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000010, A013959.

Cf. A020492, A015759, A015761, A015762, A015763, A015764, A015765, A015766, A015767, A015768, A015770, A015771, A015773, A015774, A094470.

Select[Range[2000],Divisible[DivisorSigma[11,#],EulerPhi[#]]&] (* Harvey P. Dale, Feb 07 2015 *)

Offset corrected by Amiram Eldar, Nov 02 2019

A082245 Sum of (n-1)-th powers of divisors of n.

Original entry on oeis.org

1, 3, 10, 73, 626, 8052, 117650, 2113665, 43053283, 1001953638, 25937424602, 743375541244, 23298085122482, 793811662272744, 29192932133689220, 1152956690052710401, 48661191875666868482, 2185928253847184914509

Offset: 1

Reinhard Zumkeller, May 22 2003

nonn

a(n) = t(n,n-1), t as defined in A082771;

a(1)=A000005(1), a(2)=A000203(2), a(3)=A001157(3), a(4)=A001158(4), a(5)=A001159(5), a(6)=A001160(6), a(7)=A013954(7), a(8)=A013955(8).

			a(6) = 1^5 + 2^5 + 3^5 + 6^5 = 1 + 32 + 243 + 7776 = 8052.

Seiichi Manyama, Table of n, a(n) for n = 1..387
M. Sugunamma, Certain results concerning sigma_k(n) and phi_k(n), Annales Polonici Mathematici, Vol. 8, No. 2 (1960), pp. 173-176.
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A023887, A000005, A000203, A001157, A001158, A001159, A001160, A013954, A013955, A013956, A013957, A013958
Cf. A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970
Cf. A001113, A013971, A013972.

[DivisorSigma(n-1, n): n in [1..20]]; // G. C. Greubel, Nov 02 2018

Table[Total[Divisors[n]^(n-1)], {n,18}] (* T. D. Noe, Oct 25 2006 *)
Table[DivisorSigma[n-1,n], {n,1,20}] (* G. C. Greubel, Nov 02 2018 *)

a(n) = sigma(n, n-1); \\ Michel Marcus, Nov 07 2017

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-(k*x)^k)^(1/k^2))))) \\ Seiichi Manyama, Jun 23 2019

[sigma(n,(n-1))for n in range(1,19)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^(k-1)*x^k/(1 - (k*x)^k). - Ilya Gutkovskiy, Nov 02 2018
L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)^(1/k^2)) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 23 2019
Limit_{n->oo} a(n)/A023887(n-1) = e (A001113) (Sugunamma, 1960). - Amiram Eldar, Apr 15 2021

Corrected by T. D. Noe, Oct 25 2006

A082771 Triangular array, read by rows: t(n,k) = Sum_{d|n} d^k, 0 <= k < n.

Original entry on oeis.org

1, 2, 3, 2, 4, 10, 3, 7, 21, 73, 2, 6, 26, 126, 626, 4, 12, 50, 252, 1394, 8052, 2, 8, 50, 344, 2402, 16808, 117650, 4, 15, 85, 585, 4369, 33825, 266305, 2113665, 3, 13, 91, 757, 6643, 59293, 532171, 4785157, 43053283, 4, 18, 130, 1134, 10642, 103158, 1015690, 10078254, 100390882, 1001953638

Offset: 1

Reinhard Zumkeller, May 21 2003

			From _R. J. Mathar_, Dec 06 2006 (Start):
The triangle may be extended to a rectangular array (A319278):
  1  1   1    1     1 1 1 1 1 1 1 ...
  2  3   5    9    17 33 65 129 257 513 1025 ...
  2  4  10   28    82 244 730 2188 6562 19684 59050 ...
  3  7  21   73   273 1057 4161 16513 65793 262657 1049601 ...
  2  6  26  126   626 3126 15626 78126 390626 1953126 9765626 ...
  4 12  50  252  1394 8052 47450 282252 1686434 10097892 60526250 ...
  2  8  50  344  2402 16808 117650 823544 5764802 40353608 282475250 ...
  4 15  85  585  4369 33825 266305 2113665 16843009 134480385 1074791425 ...
  3 13  91  757  6643 59293 532171 4785157 43053283 387440173 3486843451 ...
  4 18 130 1134 10642 103158 1015690 10078254 100390882 1001953638... (End)

T. D. Noe, Rows n=1..100, flattened
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000005, A000203, A001157, A001158, A001159, A001160.
Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963.
Cf. A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.

T:= (n,k)-> numtheory[sigma][k](n):
seq(seq(T(n,k), k=0..n-1), n=1..10);  # Alois P. Heinz, Oct 25 2024

T[n_, k_] := DivisorSigma[k, n];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Dec 16 2021 *)

row(n) = {my(f = factor(n)); vector(n, k, sigma(f, k-1));} \\ Amiram Eldar, May 09 2025

t(n, k) = Product(((p^((e(n, p)+1)*k))-1)/(p^k-1): n=Product(p^e(n, p): p prime)), 0<=k

t(n,0) = A000005(n), t(n,n) = A023887(n).

t(n,1) = A000203(n), n>1; t(n,2) = A001157(n), n>2; t(n,3) = A001158(n), n>3.

t(n,4) = A001159(n), n>4; t(n,5) = A001160(n), n>5; t(n,6) = A013954(n), n>6.

From R. J. Mathar, Oct 29 2006: (Start)

t(2,k) = A000051(k); t(3,k) = A034472(k); t(4,k) = A001576(k);

t(5,k) = A034474(k); t(6,k) = A034488(k); t(7,k) = A034491(k);

t(8,k) = A034496(k); t(9,k) = A034513(k); t(10,k) = A034517(k);

t(11,k) = A034524(k); t(12,k) = A034660(k). (End)

Corrected by R. J. Mathar, Dec 05 2006

A077803 Non-balanced numbers in A015769.

Original entry on oeis.org

235, 329, 470, 658, 695, 705, 799, 807, 940, 987, 1316, 1390, 1410, 1529, 1598, 1614, 1645, 1786, 1880, 1969, 1974, 2085, 2115, 2397, 2632, 2734, 2820, 3055, 3058, 3290, 3478, 3938, 3948, 4136, 4170, 4230, 4465, 4587, 4794, 4935, 5264, 5358, 5593, 5640

Offset: 1

Labos Elemer, Dec 05 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A013959, A015769, A020492, A078539.

Cf. A074866, A074868, A077801, A078544, A078549, A078550.

q[n_] := Sign[Mod[DivisorSigma[{1, 11}, n], EulerPhi[n]]] == {1, 0}; Select[Range[6000], q] (* Amiram Eldar, Apr 11 2024 *)

is(n) = {my(f = factor(n), phi = eulerphi(f)); (sigma(f) % phi) && !(sigma(f, 11) % phi);} \\ Amiram Eldar, Apr 11 2024

sigma_11(a(n)) mod phi(a(n)) = 0; sigma(a(n)) mod phi(a(n)) <> 0.

A211347 Numbers n such that n = sigma_k(m) for some k >= 1.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 24, 26, 28, 30, 31, 32, 33, 36, 38, 39, 40, 42, 44, 48, 50, 54, 56, 57, 60, 62, 63, 65, 68, 72, 73, 74, 78, 80, 82, 84, 85, 90, 91, 93, 96, 98, 102, 104, 108, 110, 112, 114, 120, 121, 122

Offset: 1

Jon Perry, Feb 05 2013

nonn

Sigma_k(n) = Sum[d|n, d^k].
Sigma_0(n) can be any positive integer and so is ignored in this sequence.
The asymptotic density of this sequence is 0 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020

			Sigma_2(4) = 1 + 4 + 16 = 21 so 21 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.
Eric W. Weisstein, MathWorld: Divisor function

Cf. A000203, A001157, A001158, A001159, A001160.
Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.
Cf. A000005.

upto[n_] := Select[Union@Flatten[{1, DivisorSigma[Range@Max[1,Floor@Log[#,n]], #] & /@ Range[2,n]}], # <= n &]; upto[122] (* Giovanni Resta, Feb 05 2013 *)

list(lim)=if(lim<3, return(if(lim<1,[],[1]))); my(v=List([1])); for(k=1,logint((lim\=1)-1,2), forfactored(m=2,sqrtnint(lim-1,k), my(t=sigma(m,k)); if(t<=lim, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Apr 09 2022

A279889 a(n) = Sum_{k=1..n-1} sigma_5(k)*sigma_5(n-k).

Original entry on oeis.org

0, 1, 66, 1577, 18218, 135550, 738236, 3207785, 11714718, 37347144, 106499470, 277489886, 668981686, 1512360404, 3228797252, 6570019945, 12793050456, 24001960051, 43483452090, 76485144056, 130752372320, 218220937122, 355664809556, 568293832670, 889969136158

Offset: 1

Seiichi Manyama, Dec 22 2016

nonn

In 1916, Ramanujan found the following identity. tau(n) = sigma_11(n) - 691/756 * (sigma_11(n) - sigma_5(n) + 252 * a(n)). This implies tau(n) == sigma_11(n) mod 691.

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory,Springer-Verlag, 1976. See p. 140, exercise 10.
Srinivasa Ramanujan, Collected papers, ed. G. H. Hardy et al., Cambridge, 1927, pp. 136-162.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
William Duke, Ramanujan and modular forms, in: K. Alladi et al., Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence, Springer Cham, 2024. See eq. (6).
D. H. Lehmer, Some functions of Ramanujan, Math. Student, Vol. 27 (1959), pp. 105-116; entire volume. See p. 111, eq. (9).
Srinivasa Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc., Vol. 22, No. 9 (1916), pp. 159-184.

Cf. Sum_{k=1..n-1} sigma_m(k)*sigma_m(n-k): A087115 (m=3), this sequence (m=5).

Cf. A000594, A001160, A013959, A027860.

a[n_] := (65 * DivisorSigma[11, n] + 691 * DivisorSigma[5, n] - 756 * RamanujanTau[n]) / 174132; Array[a, 25] (* Amiram Eldar, Jan 07 2025 *)

a(n) = sum(k=1, n-1, sigma(k, 5)*sigma(n-k, 5)) \\ Felix Fröhlich, Jan 01 2017

a(n) = {my(f = factor(n)); (65 * sigma(f, 11) + 691 * sigma(f, 5) - 756 * ramanujantau(n)) / 174132;} \\ Amiram Eldar, Jan 07 2025

A027860(n) = (sigma_11(n) - sigma_5(n) + 252*a(n))/756.

Showing 1-10 of 26 results. Next

cp's OEIS Frontend

A027860 a(n) = (-tau(n) + sigma_11(n)) / 691, where tau is Ramanujan's tau (A000594), sigma_11(n) = Sum_{ d divides n } d^11 (A013959).

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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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A008408 Theta series of Leech lattice.

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A046694 Ramanujan tau numbers mod 691 = sum of 11th power of divisors mod 691.

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A015769 Numbers k such that phi(k) | sigma_11(k).