A351267 Sum of the 4th powers of the squarefree divisors of n.

1, 17, 82, 17, 626, 1394, 2402, 17, 82, 10642, 14642, 1394, 28562, 40834, 51332, 17, 83522, 1394, 130322, 10642, 196964, 248914, 279842, 1394, 626, 485554, 82, 40834, 707282, 872644, 923522, 17, 1200644, 1419874, 1503652, 1394, 1874162, 2215474, 2342084, 10642, 2825762, 3348388

Offset

1,2

Comments

Inverse Möbius transform of n^4 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

N. J. A. Sloane, Transforms.

Formula

a(n) = Sum_{d|n} d^4 * mu(d)^2.

G.f.: Sum_{k>=1} mu(k)^2 * k^4 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022

Multiplicative with a(p^e) = 1 + p^4. - Amiram Eldar, Feb 06 2022

Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(2)) = 0.126075... . - Amiram Eldar, Nov 10 2022

Example

a(4) = 17; a(4) = Sum_{d|4} d^4 * mu(d)^2 = 1^4*1 + 2^4*1 + 4^4*0 = 17.

Mathematica

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^4); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

Prog

(PARI) a(n) = sumdiv(n, d, if (issquarefree(d), d^4)); \\ Michel Marcus, Feb 06 2022

Crossrefs

Cf. A008683 (mu), A013661, A013663.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), this sequence (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

Sequence in context: A197397 A354012 A053826 * A184982 A088687 A321560

Adjacent sequences: A351264 A351265 A351266 * A351268 A351269 A351270

Keyword

nonn,mult

Author

Wesley Ivan Hurt, Feb 05 2022

Status

approved