OFFSET
0,3
COMMENTS
Multiplicative because A013955 is. - Andrew Howroyd, Jul 25 2018
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: phi_{8, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
If p is a prime, a(p) = p^8 + p = A196288(p). - Seiichi Manyama, Feb 10 2017
a(n) = n*A013955(n) for n > 0. - Seiichi Manyama, Feb 18 2017
Sum_{k=1..n} a(k) ~ zeta(8) * n^9 / 9. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(7*e+7)-1)/(p^7-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-8). (End)
EXAMPLE
a(6) = 1^8*6^1 + 2^8*3^1 + 3^8*2^1 + 6^8*1^1 = 1693512.
MATHEMATICA
terms = 25;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E4[x]*(E2[x]*E4[x] - E6[x])/720 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[7, n], {n, 0, 24}] (* Amiram Eldar, Sep 06 2023 *)
PROG
(PARI) a(n) = if(n < 1, 0, n*sigma(n, 7)) \\ Andrew Howroyd, Jul 25 2018
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Seiichi Manyama, Feb 05 2017
STATUS
approved