A351608 a(n) = n^10 * Sum_{d^2|n} 1 / d^10.

1, 1024, 59049, 1049600, 9765625, 60466176, 282475249, 1074790400, 3486843450, 10000000000, 25937424601, 61977830400, 137858491849, 289254654976, 576650390625, 1100586418176, 2015993900449, 3570527692800, 6131066257801, 10250000000000, 16679880978201

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

Cf. A013670.

Programs

Mathematica

f[p_, e_] := p^10*(p^(10*e) - p^(10*Floor[(e - 1)/2]))/(p^10 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 13 2022 *)

PARI

a(n) = n^10*sumdiv(n, d, if (issquare(d), 1/d^5)); \\ Michel Marcus, Feb 15 2022

Formula

Multiplicative with a(p^e) = p^10*(p^(10*e) - p^(10*floor((e-1)/2)))/(p^10 - 1). - Sebastian Karlsson, Mar 03 2022

Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(12)/11 = 691*Pi^12/7023641625 = 0.090931... . - Amiram Eldar, Nov 13 2022