%I #27 Nov 14 2022 01:38:57
%S 1,8,27,72,125,216,343,576,756,1000,1331,1944,2197,2744,3375,4672,
%T 4913,6048,6859,9000,9261,10648,12167,15552,15750,17576,20412,24696,
%U 24389,27000,29791,37376,35937,39304,42875,54432,50653,54872,59319,72000,68921,74088,79507,95832
%N a(n) = n^3 * Sum_{d^2|n} 1 / d^3.
%H Seiichi Manyama, <a href="/A351601/b351601.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(p^e) = p^3*(p^(3*e) - p^(3*floor((e-1)/2)))/(p^3 - 1). - _Sebastian Karlsson_, Feb 25 2022
%F Sum_{k=1..n} a(k) ~ c * n^4, where c = zeta(5)/4 = 0.259231... . - _Amiram Eldar_, Nov 13 2022
%t f[p_, e_] := p^3*(p^(3*e) - p^(3*Floor[(e - 1)/2]))/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 13 2022 *)
%o (PARI) a(n) = n^3*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^3))); \\ _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), this sequence (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).
%Y Cf. A013663.
%K nonn,mult
%O 1,2
%A _Wesley Ivan Hurt_, Feb 14 2022