A351313 - OEIS

%I #19 Jun 21 2024 17:25:26

%S 1,1,1,16385,1,1,1,16385,4782970,1,1,16385,1,1,1,268451841,1,4782970,

%T 1,16385,1,1,1,16385,6103515626,1,4782970,16385,1,1,1,268451841,1,1,1,

%U 78368963450,1,1,1,16385,1,1,1,16385,4782970,1,1,268451841,678223072850,6103515626,1

%N Sum of the 7th powers of the square divisors of n.

%C Inverse Möbius transform of n^7 * c(n), where c(n) is the characteristic function of squares (A010052). - _Wesley Ivan Hurt_, Jun 21 2024

%H Michael De Vlieger, <a href="/A351313/b351313.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d^2|n} (d^2)^7.

%F Multiplicative with a(p) = (p^(14*(1+floor(e/2))) - 1)/(p^14 - 1). - _Amiram Eldar_, Feb 07 2022

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

%F Dirichlet g.f.: zeta(s) * zeta(2*s-14).

%F Sum_{k=1..n} a(k) ~ (zeta(15/2)/15) * n^(15/2). (End)

%F a(n) = Sum_{d|n} d^7 * c(d), where c = A010052. - _Wesley Ivan Hurt_, Jun 21 2024

%e a(16) = 268451841; a(16) = Sum_{d^2|16} (d^2)^7 = (1^2)^7 + (2^2)^7 + (4^2)^7 = 268451841.

%t f[p_, e_] := (p^(14*(1 + Floor[e/2])) - 1)/(p^14 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Feb 07 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), this sequence (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

%Y Cf. A010052.

%K nonn,easy,mult

%O 1,4

%A _Wesley Ivan Hurt_, Feb 06 2022