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a(n) = n^10 * Sum_{d^2|n} 1 / d^10.
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%I #27 Nov 14 2022 01:37:16

%S 1,1024,59049,1049600,9765625,60466176,282475249,1074790400,

%T 3486843450,10000000000,25937424601,61977830400,137858491849,

%U 289254654976,576650390625,1100586418176,2015993900449,3570527692800,6131066257801,10250000000000,16679880978201

%N a(n) = n^10 * Sum_{d^2|n} 1 / d^10.

%H Seiichi Manyama, <a href="/A351608/b351608.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p^10*(p^(10*e) - p^(10*floor((e-1)/2)))/(p^10 - 1). - _Sebastian Karlsson_, Mar 03 2022

%F Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(12)/11 = 691*Pi^12/7023641625 = 0.090931... . - _Amiram Eldar_, Nov 13 2022

%t f[p_, e_] := p^10*(p^(10*e) - p^(10*Floor[(e - 1)/2]))/(p^10 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Nov 13 2022 *)

%o (PARI) a(n) = n^10*sumdiv(n, d, if (issquare(d), 1/d^5)); \\ _Michel Marcus_, Feb 15 2022

%Y Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), this sequence (k=10).

%Y Cf. A013670.

%K nonn,mult

%O 1,2

%A _Wesley Ivan Hurt_, Feb 14 2022