OFFSET
1,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k/(1 - x^k)^2.
a(n) = n * (d(n) + sigma(n))/2.
Dirichlet g.f.: zeta(s-1) * (zeta(s-2) + zeta(s-1))/2.
a(n) = Sum_{k=1..n} k*tau(gcd(n,k)). - Ridouane Oudra, Nov 28 2019
MAPLE
with(numtheory): seq(n*(tau(n)+sigma(n))/2, n=1..30); # Ridouane Oudra, Nov 28 2019
MATHEMATICA
nmax = 55; CoefficientList[Series[Sum[k x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DirichletConvolve[j, j (j + 1)/2, j, n], {n, 1, 55}]
Table[n (DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 55}]
PROG
(PARI) a(n)=sumdiv(n, d, binomial(n/d+1, 2)*d); \\ Andrew Howroyd, Aug 14 2019
(PARI) a(n)=n*(numdiv(n) + sigma(n))/2; \\ Andrew Howroyd, Aug 14 2019
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+1, 2)*x^k/(1-x^k)^2)) \\ Seiichi Manyama, Apr 19 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 14 2019
STATUS
approved