A351607 a(n) = n^9 * Sum_{d^2|n} 1 / d^9.

1, 512, 19683, 262656, 1953125, 10077696, 40353607, 134479872, 387440172, 1000000000, 2357947691, 5169858048, 10604499373, 20661046784, 38443359375, 68853956608, 118587876497, 198369368064, 322687697779, 513000000000, 794280046581, 1207269217792, 1801152661463

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), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), this sequence (k=9), A351608 (k=10).

Cf. A013669.

Programs

Mathematica

f[p_, e_] := p^9*(p^(9*e) - p^(9*Floor[(e - 1)/2]))/(p^9 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

PARI

a(n) = n^9*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^9))); \\ Michel Marcus, Feb 15 2022

Formula

Multiplicative with a(p^e) = p^9*(p^(9*e) - p^(9*floor((e-1)/2)))/(p^9 - 1). - Sebastian Karlsson, Mar 03 2022

Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(11)/10 = 0.100049... . - Amiram Eldar, Nov 13 2022