A351605 a(n) = n^7 * Sum_{d^2|n} 1 / d^7.

1, 128, 2187, 16512, 78125, 279936, 823543, 2113536, 4785156, 10000000, 19487171, 36111744, 62748517, 105413504, 170859375, 270548992, 410338673, 612499968, 893871739, 1290000000, 1801088541, 2494357888, 3404825447, 4622303232, 6103593750, 8031810176, 10465136172

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), this sequence (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013667.

Programs

Mathematica

f[p_, e_] := p^7*(p^(7*e) - p^(7*Floor[(e - 1)/2]))/(p^7 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

PARI

a(n) = n^7*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^7))); \\ Michel Marcus, Feb 15 2022

Formula

Multiplicative with a(p^e) = p^7*(p^(7*e) - p^(7*floor((e-1)/2)))/(p^7 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^8, where c = zeta(9)/8 = 0.125251... . - Amiram Eldar, Nov 13 2022