A351606 a(n) = n^8 * Sum_{d^2|n} 1 / d^8.

1, 256, 6561, 65792, 390625, 1679616, 5764801, 16842752, 43053282, 100000000, 214358881, 431661312, 815730721, 1475789056, 2562890625, 4311810048, 6975757441, 11021640192, 16983563041, 25700000000, 37822859361, 54875873536, 78310985281, 110505295872, 152588281250

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

Cf. A013668.

Programs

Mathematica

f[p_, e_] := p^8*(p^(8*e) - p^(8*Floor[(e - 1)/2]))/(p^8 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

PARI

a(n) = n^8*sumdiv(n, d, if (issquare(d), 1/d^4)); \\ Michel Marcus, Feb 15 2022

Formula

Multiplicative with a(p^e) = p^8*(p^(8*e) - p^(8*floor((e-1)/2)))/(p^8 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(10)/9 = Pi^10/841995 = 0.1112216... . - Amiram Eldar, Nov 13 2022