OFFSET
0,3
COMMENTS
Multiplicative because A013957 is. - Andrew Howroyd, Jul 25 2018
D. H. Lehmer shows that a(n) == tau(n) (mod 7) for n > 0, where tau is Ramanujan's tau function (A000594). Furthermore, if n == 3, 5, 6 (mod 7) then a(n) == tau(n) (mod 49). See the Wikipedia link below. - Jianing Song, Aug 12 2020
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
Wikipedia, Congruences for the tau function.
FORMULA
G.f.: phi_{10, 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^10 + p = A196292(p).
a(n) = n*A013957(n) for n > 0, where A013957(n) is sigma_9(n), the sum of the 9th powers of the divisors of n. - Seiichi Manyama, Feb 18 2017
Multiplicative with a(p^e) = p^e*(p^(9*(e+1))-1)/(p^9-1). - Jianing Song, Aug 12 2020
From Amiram Eldar, Oct 30 2023: (Start)
Dirichlet g.f.: zeta(s-1)*zeta(s-10).
Sum_{k=1..n} a(k) ~ zeta(10) * n^11 / 11. (End)
EXAMPLE
a(6) = 1^10*6^1 + 2^10*3^1 + 3^10*2^1 + 6^10*1^1 = 60587352.
MATHEMATICA
Table[If[n>0, n * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)
PROG
(PARI) for(n=0, 20, print1(if(n==0, 0, n * sigma(n, 9)), ", ")) \\ Indranil Ghosh, Mar 12 2017
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Seiichi Manyama, Feb 10 2017
STATUS
approved