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A351310
Sum of the 5th powers of the square divisors of n.
11
1, 1, 1, 1025, 1, 1, 1, 1025, 59050, 1, 1, 1025, 1, 1, 1, 1049601, 1, 59050, 1, 1025, 1, 1, 1, 1025, 9765626, 1, 59050, 1025, 1, 1, 1, 1049601, 1, 1, 1, 60526250, 1, 1, 1, 1025, 1, 1, 1, 1025, 59050, 1, 1, 1049601, 282475250, 9765626, 1, 1025, 1, 59050, 1, 1025, 1, 1
OFFSET
1,4
COMMENTS
Inverse Möbius transform of n^5 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024
LINKS
FORMULA
a(n) = Sum_{d^2|n} (d^2)^5.
Multiplicative with a(p) = (p^(10*(1+floor(e/2))) - 1)/(p^10 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-10).
Sum_{k=1..n} a(k) ~ (zeta(11/2)/11) * n^(11/2). (End)
a(n) = Sum_{d|n} d^5 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024
EXAMPLE
a(16) = 1049601; a(16) = Sum_{d^2|16} (d^2)^5 = (1^2)^5 + (2^2)^5 + (4^2)^5 = 1049601.
MATHEMATICA
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 *)
CROSSREFS
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).
Cf. A010052.
Sequence in context: A017554 A031147 A087932 * A045035 A060948 A171385
KEYWORD
nonn,easy,mult
AUTHOR
Wesley Ivan Hurt, Feb 06 2022
STATUS
approved