A351602 a(n) = n^4 * Sum_{d^2|n} 1 / d^4.

1, 16, 81, 272, 625, 1296, 2401, 4352, 6642, 10000, 14641, 22032, 28561, 38416, 50625, 69888, 83521, 106272, 130321, 170000, 194481, 234256, 279841, 352512, 391250, 456976, 538002, 653072, 707281, 810000, 923521, 1118208, 1185921, 1336336, 1500625, 1806624, 1874161, 2085136

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), this sequence (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013664.

Programs

Mathematica

f[p_, e_] := p^4*(p^(4*e) - p^(4*Floor[(e - 1)/2]))/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 13 2022 *)

PARI

a(n) = n^4*sumdiv(n, d, if (issquare(d), 1/d^2)); \\ Michel Marcus, Feb 15 2022

Formula

Multiplicative with a(p^e) = p^4*(p^(4*e) - p^(4*floor((e-1)/2)))/(p^4 - 1). - Sebastian Karlsson, Feb 25 2022

Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(6)/5 = Pi^6/4725 = 0.203468... . - Amiram Eldar, Nov 13 2022