A351604 a(n) = n^6 * Sum_{d^2|n} 1 / d^6.

1, 64, 729, 4160, 15625, 46656, 117649, 266240, 532170, 1000000, 1771561, 3032640, 4826809, 7529536, 11390625, 17043456, 24137569, 34058880, 47045881, 65000000, 85766121, 113379904, 148035889, 194088960, 244156250, 308915776, 387951930, 489419840, 594823321, 729000000

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), this sequence (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013666.

Programs

Mathematica

f[p_, e_] := p^6*(p^(6*e) - p^(6*Floor[(e - 1)/2]))/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

PARI

a(n) = n^6*sumdiv(n, d, if (issquare(d), 1/d^3)); \\ Michel Marcus, Feb 15 2022

Formula

Multiplicative with a(p^e) = p^6*(p^(6*e) - p^(6*floor((e-1)/2)))/(p^6 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(8)/7 = Pi^8/66150 = 0.143439... . - Amiram Eldar, Nov 13 2022