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).

Triangle A144736: row sums = sigma(n), right border = phi(n), left border = d(n). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2008]

Regarding Euler's recurring sequence for sigma(n). Let s=sigma, then Euler states [Young, p.361]: "...I say that the value of s(n) can always be combined from some of the preceding as prescribed by the following formula: s(n) = s(n-1) + s(n-2) - s(n-5) - s(n-7) + s(n-12) + s(n-15) - s(n-22) - s(n-26) + ..." [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 05 2008]

Prefaced with a zero: (0, 1, 3, 4, 7,...) = A147843 convolved with the partition numbers, A000041. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2008]

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.

J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Acta Cryst. A48 (1992), 500-508. [From N. J. A. Sloane, Mar 14 2009]

J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices II, Acta Cryst. A49 (1993), 293-300. [From 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, 156163. [See Table 1]. [From N. J. A. Sloane, (njas(AT)research.att.com), Feb 23 2009]

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).

Robert M. Young, "Excursions in Calculus", The Mathematical Association of America, 1992 p.361 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 05 2008]

Daniel Forgues, Table of n, a(n) for n=1..100000

Walter Nissen, Abundancy : Some Resources

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].

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)gmail.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)gmail.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)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

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 (jvospost3(AT)gmail.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 */

(MuPad) numlib::sigma(n)$ n=1..81 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 13 2008

(PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)

N=17; default(seriesprecision, N); x=z+O(z^(N+1))

c=sum(j=1, N, j*x^j); \\ log case

s=-log(prod(j=1, N, 1-x^j)); \\ A000203 sum of divisors

s=serconvol(s, c)

v=Vec(s)

(Other) sage: [sigma(n, 1)for n in xrange(1, 71)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]

(PARI) max_n = 30; ser = - sum(k=1, max_n, log(1-x^k)) a(n) = polcoeff(ser, n)*n [From Gottfried Helms (helms(AT)uni-kassel.de), Aug 10 2009]

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.

A144736 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2008]

A147843 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2008]

Equals row sums of triangle A158951 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 31 2009]

Equals row sums of triangle A158902 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 29 2009]

Sequence in context: A097863 A097012 A143348 this_sequence A003979 A084250 A090128

Adjacent sequences: A000200 A000201 A000202 this_sequence A000204 A000205 A000206

easy,core,nonn,nice,mult

N. J. A. Sloane (njas(AT)research.att.com).