A351267 - OEIS

%I #27 Jun 08 2023 00:23:44

%S 1,17,82,17,626,1394,2402,17,82,10642,14642,1394,28562,40834,51332,17,

%T 83522,1394,130322,10642,196964,248914,279842,1394,626,485554,82,

%U 40834,707282,872644,923522,17,1200644,1419874,1503652,1394,1874162,2215474,2342084,10642,2825762,3348388

%N Sum of the 4th powers of the squarefree divisors of n.

%C Inverse Möbius transform of n^4 * mu(n)^2. - _Wesley Ivan Hurt_, Jun 08 2023

%H Seiichi Manyama, <a href="/A351267/b351267.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^4 * mu(d)^2.

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

%F Multiplicative with a(p^e) = 1 + p^4. - _Amiram Eldar_, Feb 06 2022

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

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

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

%o (PARI) a(n) = sumdiv(n, d, if (issquarefree(d), d^4)); \\ _Michel Marcus_, Feb 06 2022

%Y Cf. A008683 (mu), A013661, A013663.

%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), this sequence (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

%K nonn,mult

%O 1,2

%A _Wesley Ivan Hurt_, Feb 05 2022