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A351314
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Sum of the 8th powers of the square divisors of n.
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11
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1, 1, 1, 65537, 1, 1, 1, 65537, 43046722, 1, 1, 65537, 1, 1, 1, 4295032833, 1, 43046722, 1, 65537, 1, 1, 1, 65537, 152587890626, 1, 43046722, 65537, 1, 1, 1, 4295032833, 1, 1, 1, 2821153019714, 1, 1, 1, 65537, 1, 1, 1, 65537, 43046722, 1, 1, 4295032833, 33232930569602, 152587890626
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OFFSET
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1,4
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LINKS
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FORMULA
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a(n) = Sum_{d^2|n} (d^2)^8.
Multiplicative with a(p) = (p^(16*(1+floor(e/2))) - 1)/(p^16 - 1). - Amiram Eldar, Feb 07 2022
Dirichlet g.f.: zeta(s) * zeta(2*s-16).
Sum_{k=1..n} a(k) ~ (zeta(17/2)/17) * n^(17/2). (End)
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EXAMPLE
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a(16) = 4295032833; a(16) = Sum_{d^2|16} (d^2)^8 = (1^2)^8 + (2^2)^8 + (4^2)^8 = 4295032833.
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MATHEMATICA
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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 *)
Table[Total[Select[Divisors[n], IntegerQ[Sqrt[#]]&]^8], {n, 80}] (* Harvey P. Dale, Feb 13 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^16*x^k^2/(1-x^k^2))) \\ Seiichi Manyama, Feb 12 2022
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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), 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).
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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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