A320940 a(n) = Sum_{d|n} d*sigma_n(d).

1, 11, 85, 1127, 15631, 287021, 5764809, 135007759, 3487020610, 100146496681, 3138428376733, 107032667155169, 3937376385699303, 155582338242604221, 6568408966322733475, 295154660699054931999, 14063084452067724991027, 708239400347943609329270

Offset: 1

Author

Ilya Gutkovskiy, Oct 28 2018

Keywords

• nonn

Examples

			a(6) = 1*sigma_6(1)+2*sigma_6(2)+3*sigma_6(3)+6*sigma_6(6) = 1+2*65+3*730+6*47450 = 287021.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..385
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000203, A001001, A027847, A027848, A319647, A321141.

Programs

Magma

[&+[d*DivisorSigma(n,d):d in Divisors(n)]:n in [1..18]]; // Marius A. Burtea, Feb 15 2020

Maple

with(numtheory): seq(coeff(series(n*(-log(mul((1-x^k)^sigma[n](k),k=1..n))),x,n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 28 2018

Mathematica

Table[Sum[d DivisorSigma[n, d], {d, Divisors[n]}] , {n, 18}]
Table[n SeriesCoefficient[-Log[Product[(1 - x^k)^DivisorSigma[n, k], {k, 1, n}]], {x, 0, n}], {n, 18}]

PARI

a(n) = sumdiv(n, d, d*sigma(d, n)); \\ Michel Marcus, Oct 28 2018

Python

from sympy import divisor_sigma, divisors
def A320940(n):
    return sum(divisor_sigma(d)*(n//d)**(n+1) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

a(n) = n * [x^n] -log(Product_{k>=1} (1 - x^k)^sigma_n(k)).

a(n) = Sum_{d|n} d^(n+1)*sigma_1(n/d).

a(n) ~ n^(n+1). - Vaclav Kotesovec, Feb 16 2020