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

Offset: 1

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

• nonn
• easy
• mult

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

Examples

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

Links

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

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.

Programs

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 *)

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