A351315 - OEIS

A351315

Sum of the 9th powers of the square divisors of n.

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

1,4

COMMENTS

Inverse Möbius transform of n^9 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

FORMULA

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

From Amiram Eldar, Sep 20 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-18).

Sum_{k=1..n} a(k) ~ (zeta(19/2)/19) * n^(19/2). (End)

a(n) = Sum_{d|n} d^9 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024

a(n) = Sum_{d|n} lambda(d)*d^9*sigma_9(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

EXAMPLE

a(16) = 68719738881; a(16) = Sum_{d^2|16} (d^2)^9 = (1^2)^9 + (2^2)^9 + (4^2)^9 = 68719738881.

MATHEMATICA

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 *)

snp[n_]:=Total[Select[Divisors[n], IntegerQ[Sqrt[#]]&]^9]; Array[snp, 50] (* Harvey P. Dale, May 25 2025 *)

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), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), this sequence (k=9), A351316 (k=10).

Cf. A010052, A008836, A013957.

Sequence in context: A017445 A017577 A051441 * A352984 A017699 A013966

Adjacent sequences: A351312 A351313 A351314 * A351316 A351317 A351318

KEYWORD

nonn,easy,mult

AUTHOR

Wesley Ivan Hurt, Feb 06 2022

STATUS

approved