A351314 - OEIS

%I #23 Jun 21 2024 17:28:44

%S 1,1,1,65537,1,1,1,65537,43046722,1,1,65537,1,1,1,4295032833,1,

%T 43046722,1,65537,1,1,1,65537,152587890626,1,43046722,65537,1,1,1,

%U 4295032833,1,1,1,2821153019714,1,1,1,65537,1,1,1,65537,43046722,1,1,4295032833,33232930569602,152587890626

%N Sum of the 8th powers of the square divisors of n.

%C Inverse Möbius transform of n^8 * c(n), where c(n) is the characteristic function of squares (A010052). - _Wesley Ivan Hurt_, Jun 21 2024

%H Seiichi Manyama, <a href="/A351314/b351314.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d^2|n} (d^2)^8.

%F Multiplicative with a(p) = (p^(16*(1+floor(e/2))) - 1)/(p^16 - 1). - _Amiram Eldar_, Feb 07 2022

%F G.f.: Sum_{k>0} k^16*x^(k^2)/(1-x^(k^2)). - _Seiichi Manyama_, Feb 12 2022

%F From _Amiram Eldar_, Sep 20 2023: (Start)

%F Dirichlet g.f.: zeta(s) * zeta(2*s-16).

%F Sum_{k=1..n} a(k) ~ (zeta(17/2)/17) * n^(17/2). (End)

%F a(n) = Sum_{d|n} d^8 * c(d), where c = A010052. - _Wesley Ivan Hurt_, Jun 21 2024

%e a(16) = 4295032833; a(16) = Sum_{d^2|16} (d^2)^8 = (1^2)^8 + (2^2)^8 + (4^2)^8 = 4295032833.

%t f[p_, e_] := (p^(16*(1 + Floor[e/2])) - 1)/(p^16 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Feb 07 2022 *)

%t Table[Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]^8],{n,80}] (* _Harvey P. Dale_, Feb 13 2022 *)

%o (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^16*x^k^2/(1-x^k^2))) \\ _Seiichi Manyama_, Feb 12 2022

%Y Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), this sequence (k=8), A351315 (k=9), A351316 (k=10).

%Y Cf. A010052.

%K nonn,easy,mult

%O 1,4

%A _Wesley Ivan Hurt_, Feb 06 2022