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A351307
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Sum of the squares of the square divisors of n.
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12
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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, 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, 1, 17, 1, 1, 1, 17, 1, 1, 82, 4369, 1, 1, 1, 17, 1, 1, 1, 1394, 1, 1, 626, 17, 1, 1
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OFFSET
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1,4
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COMMENTS
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Inverse Möbius transform of n^2 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024
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LINKS
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FORMULA
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a(n) = Sum_{d^2|n} (d^2)^2.
Multiplicative with a(p) = (p^(4*(1+floor(e/2))) - 1)/(p^4 - 1). - Amiram Eldar, Feb 07 2022
Dirichlet g.f.: zeta(s) * zeta(2*s-4).
Sum_{k=1..n} a(k) ~ (zeta(5/2)/5) * n^(5/2). (End)
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EXAMPLE
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a(16) = 273; a(16) = Sum_{d^2|16} (d^2)^2 = (1^2)^2 + (2^2)^2 + (4^2)^2 = 273.
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MATHEMATICA
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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 *)
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PROG
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(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
(Python)
from math import prod
from sympy import factorint
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
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CROSSREFS
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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).
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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