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

Author

Benoit Cloitre, Apr 20 2003

Keywords

• nonn

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

Crossrefs

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

Programs

Mathematica

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

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

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

Extensions

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