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%I #27 Jun 08 2023 00:26:14
%S 1,1025,59050,1025,9765626,60526250,282475250,1025,59050,10009766650,
%T 25937424602,60526250,137858491850,289537131250,576660215300,1025,
%U 2015993900450,60526250,6131066257802,10009766650,16680163512500,26585860217050,41426511213650,60526250
%N Sum of the 10th powers of the squarefree divisors of n.
%C Inverse Möbius transform of n^10 * mu(n)^2. - _Wesley Ivan Hurt_, Jun 08 2023
%H Seiichi Manyama, <a href="/A351273/b351273.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^10 * mu(d)^2.
%F Multiplicative with a(p^e) = 1 + p^10. - _Amiram Eldar_, Feb 06 2022
%F G.f.: Sum_{k>=1} mu(k)^2 * k^10 * x^k / (1 - x^k). - _Ilya Gutkovskiy_, Feb 06 2022
%F Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(11)/(11*zeta(2)) = 0.0552934... . - _Amiram Eldar_, Nov 10 2022
%e a(4) = 1025; a(4) = Sum_{d|4} d^10 * mu(d)^2 = 1^10*1 + 2^10*1 + 4^10*0 = 1025.
%t a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^10); Array[a, 100] (* _Amiram Eldar_, Feb 06 2022 *)
%t Table[Total[Select[Divisors[n],SquareFreeQ]^10],{n,25}] (* _Harvey P. Dale_, Nov 20 2022 *)
%Y Cf. A008683 (mu), A013661, A013669.
%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), A351271 (k=8), A351272 (k=9), this sequence (k=10).
%K nonn,mult
%O 1,2
%A _Wesley Ivan Hurt_, Feb 05 2022
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