A081307 a(n) = (n+1)*tau(n) - sigma(n).

1, 3, 4, 8, 6, 16, 8, 21, 17, 26, 12, 50, 14, 36, 40, 54, 18, 75, 20, 84, 56, 56, 24, 140, 47, 66, 72, 118, 30, 176, 32, 135, 88, 86, 96, 242, 38, 96, 104, 238, 42, 248, 44, 186, 198, 116, 48, 366, 93, 213, 136, 220

Offset

1,2

Comments

Old name was: Sum_{k=1..n} Sum_{m=1..k} 1/(1-x^m).

Number of positive integer pairs (s,t) with s <= t <= n, such that s|n. For example, when n = 6, the 16 pairs are (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,2), (2,3), (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6), (6,6). - Wesley Ivan Hurt, Nov 15 2021

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Formula

Sum_{k=1..n} Sum_{m=1..k} 1/(1-x^m).

a(n) = Sum_{k=1..n} k*A113998(n,k). - Philippe Deléham, Feb 03 2007

Mathematica

Table[(n + 1) DivisorSigma[0, n] - DivisorSigma[1, n], {n, 100}] (* Wesley Ivan Hurt, Nov 15 2021 *)

Prog

(PARI) a(n)=if(n<1, 0, polcoeff(sum(k=1, n, sum(l=1, k, 1/(1-x^l)), x*O(x^n)), n))
(PARI) a(n)=sum(j=1, n, sum(k=1, j, n%k==0)) \\ Hugo Pfoertner, Jul 09 2025

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A094471, A113998.

Sequence in context: A347228 A330575 A079787 * A344225 A081543 A328876

Adjacent sequences: A081304 A081305 A081306 * A081308 A081309 A081310

Keyword

nonn

Author

Benoit Cloitre, Apr 20 2003

Extensions

Name changed by Wesley Ivan Hurt, Nov 16 2021 using formula from Vladeta Jovovic, Jan 22 2005

Status

approved