A351601 a(n) = n^3 * Sum_{d^2|n} 1 / d^3.

1, 8, 27, 72, 125, 216, 343, 576, 756, 1000, 1331, 1944, 2197, 2744, 3375, 4672, 4913, 6048, 6859, 9000, 9261, 10648, 12167, 15552, 15750, 17576, 20412, 24696, 24389, 27000, 29791, 37376, 35937, 39304, 42875, 54432, 50653, 54872, 59319, 72000, 68921, 74088, 79507, 95832

Offset: 1

Author

Wesley Ivan Hurt, Feb 14 2022

Keywords

• nonn
• mult

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

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).

Cf. A013663.

Programs

Mathematica

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 *)

PARI

a(n) = n^3*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^3))); \\ Michel Marcus, Feb 15 2022

Formula

Multiplicative with a(p^e) = p^3*(p^(3*e) - p^(3*floor((e-1)/2)))/(p^3 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^4, where c = zeta(5)/4 = 0.259231... . - Amiram Eldar, Nov 13 2022