A351600 a(n) = n^2 * Sum_{d^2|n} 1 / d^2.

1, 4, 9, 20, 25, 36, 49, 80, 90, 100, 121, 180, 169, 196, 225, 336, 289, 360, 361, 500, 441, 484, 529, 720, 650, 676, 810, 980, 841, 900, 961, 1344, 1089, 1156, 1225, 1800, 1369, 1444, 1521, 2000, 1681, 1764, 1849, 2420, 2250, 2116, 2209, 3024, 2450, 2600, 2601, 3380, 2809

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), this sequence (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Cf. A013662, A076752.

Programs

Mathematica

f[p_, e_] := p^2*(p^(2*e) - p^(2*Floor[(e - 1)/2]))/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

PARI

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

Formula

G.f.: Sum_{k>=1} k^2 * x^(k^2) * (1 + x^(k^2)) / (1 - x^(k^2))^3. - Ilya Gutkovskiy, Feb 21 2022

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

Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(4)/3 = Pi^4/270 = 0.360774... . - Amiram Eldar, Nov 13 2022