1,2

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. - comment from 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) = number of sublattices of index n in a generic 2-dimensional lattice - Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001

The sublattices of index n are in one-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} = sigma(n), which is A000203. 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 (pcmusume(AT)m11.alpha-net.ne.jp), 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).

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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 2nd formula.

J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.

M. J. Grady, A group theoretic approach to a famous partition formula, Amer. Math. Monthly, 112 (No. 7, 2005), 645-651.

Ross Honsberger, "Mathematical Gems, Number One," The Dolciani Mathematical Expositions, Published and Distributed by The Mathematical Association of America, page 116.

M. Krasner, Le nombre des surcorps primitifs d'un degre donne et le nombre des surcorps metagaloisiens d'un degre donne d'un corp de nombre p-adique. Comptes Redus Hebdomadaires, Acadmie des Science, 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.

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.

G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ Press 1954, page 92.

N. J. A. Sloane, Table of n, sigma(n) for n = 1..20000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

M. Baake and U. Grimm, Quasicrystalline combinatorics

H. Bottomley, Illustration of initial terms

C. K. Caldwell, The Prime Glossary, sigma function

L. Euler, Observatio de summis divisorum

L. Euler, An observation on the sums of divisors

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.

M. Maia and M. Mendez, On the arithmetic product of combinatorial species

K. Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)

Jon Perry, More Partition Functions

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to sublattices

Index entries for "core" sequences

Dirichlet convolution of phi(n) and tau(n), i.e. a(n)=Sum_{d|n} phi(n/d)*tau(d), cf. A000010, A000005.

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.

sigma(n) is odd iff n is a square or twice a square. - Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 03 2001

sigma[n]=sigma[n*p(n)]-p(n)*sigma[n] - Labos E. (labos(AT)ana.sote.hu), Aug 14 2003

a(n) = n*A000041(n) - sum{i=1, n-1, a(i)*A000041(n-i)} - Jon Perry (perry(AT)globalnet.co.uk), Sep 11 2003

a(n) = -A010815(n)*n - Sum(A010815(k)*a(n-k): 1<=k<n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), 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 (reinhard.zumkeller(AT)lhsystems.com), Nov 17 2004

Recurrence: sigma(1) = 1 sigma(n) = 12*Sum[(5*k*(n-k)-n^2)*sigma(k)*sigma(n-k), k=1..(n-1)]/((n^2)*(n-1)) if n>1 - Dominique Giard (dominique.giard(AT)caramail.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

For odd n, A000203(n) = A000593(n) sum of odd divisors of n. For even n, A000203(n) = A000593(n) + A074400(n/2) where A074400 is sum of the even divisors of 2n. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 26 2006

Equals A051731 * [1,2,3,...]; the inverse Moebius transform of the natural numbers. Equals row sums of A127093 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 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 (qntmpkt(AT)yahoo.com), Oct 31 2007

Row sums of triangle A134838 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 12 2007

a(n) = A054785(2*n) - A000593(2*n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 23 2008

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.

with(numtheory): A000203 := n->sigma(n);

Table[ DivisorSigma[1, n], {n, 1, 100} ]

(MAGMA) [ SumOfDivisors(n) : n in [1..40]];

(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/(x^k-1)^2, x*O(x^n)), n)) /* Michael Somos Jan 29 2005 */

Cf. A001157, A001158, A001160, A001065, A002192, A001001, A001615 (primitive sublattices).

See A034885, A002093 for records. Bisections give A008438, A062731.

Cf. A039653, A088580, A074400, A029416, A083728, A006352, A002659, A083238.

Cf. A000593, A074400, A050449, A050452.

Cf. A051731, A127093.

Cf. A134838.

Adjacent sequences: A000200 A000201 A000202 this_sequence A000204 A000205 A000206

Sequence in context: A074847 A097863 A097012 this_sequence A003979 A084250 A090128

easy,core,nonn,nice,mult,new

njas