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A351315
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Sum of the 9th powers of the square divisors of n.
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11
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1, 1, 1, 262145, 1, 1, 1, 262145, 387420490, 1, 1, 262145, 1, 1, 1, 68719738881, 1, 387420490, 1, 262145, 1, 1, 1, 262145, 3814697265626, 1, 387420490, 262145, 1, 1, 1, 68719738881, 1, 1, 1, 101560344351050, 1, 1, 1, 262145, 1, 1, 1, 262145, 387420490, 1, 1, 68719738881
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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^9 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024
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LINKS
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FORMULA
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a(n) = Sum_{d^2|n} (d^2)^9.
Multiplicative with a(p) = (p^(18*(1+floor(e/2))) - 1)/(p^18 - 1). - Amiram Eldar, Feb 07 2022
Dirichlet g.f.: zeta(s) * zeta(2*s-18).
Sum_{k=1..n} a(k) ~ (zeta(19/2)/19) * n^(19/2). (End)
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EXAMPLE
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a(16) = 68719738881; a(16) = Sum_{d^2|16} (d^2)^9 = (1^2)^9 + (2^2)^9 + (4^2)^9 = 68719738881.
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MATHEMATICA
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f[p_, e_] := (p^(18*(1 + Floor[e/2])) - 1)/(p^18 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 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), A351314 (k=8), this sequence (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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