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Sum of the squares of the square divisors of n.
12

%I #28 Jul 11 2024 16:53:27

%S 1,1,1,17,1,1,1,17,82,1,1,17,1,1,1,273,1,82,1,17,1,1,1,17,626,1,82,17,

%T 1,1,1,273,1,1,1,1394,1,1,1,17,1,1,1,17,82,1,1,273,2402,626,1,17,1,82,

%U 1,17,1,1,1,17,1,1,82,4369,1,1,1,17,1,1,1,1394,1,1,626,17,1,1

%N Sum of the squares of the square divisors of n.

%C Inverse Möbius transform of n^2 * c(n), where c(n) is the characteristic function of squares (A010052). - _Wesley Ivan Hurt_, Jun 20 2024

%H Michael De Vlieger, <a href="/A351307/b351307.txt">Table of n, a(n) for n = 1..10000</a>

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

%F Multiplicative with a(p) = (p^(4*(1+floor(e/2))) - 1)/(p^4 - 1). - _Amiram Eldar_, Feb 07 2022

%F G.f.: Sum_{k>0} k^4*x^(k^2)/(1-x^(k^2)). - _Seiichi Manyama_, Feb 12 2022

%F From _Amiram Eldar_, Sep 19 2023: (Start)

%F Dirichlet g.f.: zeta(s) * zeta(2*s-4).

%F Sum_{k=1..n} a(k) ~ (zeta(5/2)/5) * n^(5/2). (End)

%F a(n) = Sum_{d|n} d^2 * c(d), where c = A010052. - _Wesley Ivan Hurt_, Jun 20 2024

%e a(16) = 273; a(16) = Sum_{d^2|16} (d^2)^2 = (1^2)^2 + (2^2)^2 + (4^2)^2 = 273.

%t f[p_, e_] := (p^(4*(1 + Floor[e/2])) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Feb 07 2022 *)

%o (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^4*x^k^2/(1-x^k^2))) \\ _Seiichi Manyama_, Feb 12 2022

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A351307(n): return prod((p**(4+((e&-2)<<1))-1)//(p**4-1) for p,e in factorint(n).items()) # _Chai Wah Wu_, Jul 11 2024

%Y Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), this sequence (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

%Y Cf. A010052, A247041 (zeta(5/2)).

%K nonn,easy,mult

%O 1,4

%A _Wesley Ivan Hurt_, Feb 06 2022