A351310 - OEIS

%I #20 Jun 21 2024 17:19:37

%S 1,1,1,1025,1,1,1,1025,59050,1,1,1025,1,1,1,1049601,1,59050,1,1025,1,

%T 1,1,1025,9765626,1,59050,1025,1,1,1,1049601,1,1,1,60526250,1,1,1,

%U 1025,1,1,1,1025,59050,1,1,1049601,282475250,9765626,1,1025,1,59050,1,1025,1,1

%N Sum of the 5th powers of the square divisors of n.

%C Inverse Möbius transform of n^5 * c(n), where c(n) is the characteristic function of squares (A010052). - _Wesley Ivan Hurt_, Jun 21 2024

%H Michael De Vlieger, <a href="/A351310/b351310.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d^2|n} (d^2)^5.

%F Multiplicative with a(p) = (p^(10*(1+floor(e/2))) - 1)/(p^10 - 1). - _Amiram Eldar_, Feb 07 2022

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

%F Dirichlet g.f.: zeta(s) * zeta(2*s-10).

%F Sum_{k=1..n} a(k) ~ (zeta(11/2)/11) * n^(11/2). (End)

%F a(n) = Sum_{d|n} d^5 * c(d), where c = A010052. - _Wesley Ivan Hurt_, Jun 21 2024

%e a(16) = 1049601; a(16) = Sum_{d^2|16} (d^2)^5 = (1^2)^5 + (2^2)^5 + (4^2)^5 = 1049601.

%t f[p_, e_] := (p^(10*(1 + Floor[e/2])) - 1)/(p^10 - 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), this sequence (k=5), A351311 (k=6), A351313 (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