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
Examples
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.
References
- 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.
Links
- 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
Crossrefs
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).
Programs
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GAP
A000203:=List([1..10^2],n->Sigma(n)); # Muniru A Asiru, Oct 01 2017
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Haskell
a000203 n = product $ zipWith (\p e -> (p^(e+1)-1) `div` (p-1)) (a027748_row n) (a124010_row n) -- Reinhard Zumkeller, May 07 2012
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Magma
[SumOfDivisors(n): n in [1..70]];
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Magma
[DivisorSigma(1,n): n in [1..70]]; // Bruno Berselli, Sep 09 2015
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Maple
with(numtheory): A000203 := n->sigma(n); seq(A000203(n), n=1..100);
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Mathematica
Table[ DivisorSigma[1, n], {n, 100}] a[ n_] := SeriesCoefficient[ QPolyGamma[ 1, 1, q] / Log[q]^2, {q, 0, n}]; (* Michael Somos, Apr 25 2013 *) -
Maxima
makelist(divsum(n),n,1,1000); /* Emanuele Munarini, Mar 26 2011 */
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MuPAD
numlib::sigma(n)$ n=1..81 // Zerinvary Lajos, May 13 2008
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PARI
{a(n) = if( n<1, 0, sigma(n))}; -
PARI
{a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) /(1 - p*X))[n])}; -
PARI
{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 */ -
PARI
max_n = 30; ser = - sum(k=1,max_n,log(1-x^k)); a(n) = polcoeff(ser,n)*n \\ Gottfried Helms, Aug 10 2009
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Python
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
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Python
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
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SageMath
[sigma(n, 1) for n in range(1, 71)] # Zerinvary Lajos, Jun 04 2009
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Scheme
(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
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Scheme
(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
Formula
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) = 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) = n*Sum_{k=1..n} A060642(n,k)/k*(-1)^(k+1). - Vladimir Kruchinin, Aug 10 2010
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) = 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) = Sum{i=1..n} Sum{j=1..i} cos((2*Pi*n*j)/i). - Michel Lagneau, Oct 14 2015
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) = 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(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
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