A351603 a(n) = n^5 * Sum_{d^2|n} 1 / d^5.

1, 32, 243, 1056, 3125, 7776, 16807, 33792, 59292, 100000, 161051, 256608, 371293, 537824, 759375, 1082368, 1419857, 1897344, 2476099, 3300000, 4084101, 5153632, 6436343, 8211456, 9768750, 11881376, 14407956, 17748192, 20511149, 24300000, 28629151, 34635776, 39135393

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), this sequence (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013665.

Programs

Mathematica

f[p_, e_] := p^5*(p^(5*e) - p^(5*Floor[(e - 1)/2]))/(p^5 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 13 2022 *)

PARI

a(n) = n^5*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^5))); \\ Michel Marcus, Feb 15 2022

Formula

Multiplicative with a(p^e) = p^5*(p^(5*e) - p^(5*floor((e-1)/2)))/(p^5 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^6, where c = zeta(7)/6 = 0.168058... . - Amiram Eldar, Nov 13 2022