%I #22 Jun 08 2023 00:25:29

%S 1,257,6562,257,390626,1686434,5764802,257,6562,100390882,214358882,

%T 1686434,815730722,1481554114,2563287812,257,6975757442,1686434,

%U 16983563042,100390882,37828630724,55090232674,78310985282,1686434,390626,209642795554,6562,1481554114,500246412962

%N Sum of the 8th powers of the squarefree divisors of n.

%C Inverse Möbius transform of n^8 * mu(n)^2. - _Wesley Ivan Hurt_, Jun 08 2023

%H Seiichi Manyama, <a href="/A351271/b351271.txt">Table of n, a(n) for n = 1..10000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%F a(n) = Sum_{d|n} d^8 * mu(d)^2.

%F Multiplicative with a(p^e) = 1 + p^8. - _Amiram Eldar_, Feb 06 2022

%F G.f.: Sum_{k>=1} mu(k)^2 * k^8 * x^k / (1 - x^k). - _Ilya Gutkovskiy_, Feb 06 2022

%F Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/(9*zeta(2)) = 0.0676831... . - _Amiram Eldar_, Nov 10 2022

%e a(4) = 257; a(4) = Sum_{d|4} d^8 * mu(d)^2 = 1^8*1 + 2^8*1 + 4^8*0 = 257.

%t a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^8); Array[a, 100] (* _Amiram Eldar_, Feb 06 2022 *)

%Y Cf. A008683 (mu), A013661, A013667.

%Y 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), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), this sequence (k=8), A351272 (k=9), A351273 (k=10).

%K nonn,mult

%O 1,2

%A _Wesley Ivan Hurt_, Feb 05 2022