A254520 Möbius transform of A034676.

1, 4, 9, 12, 25, 36, 49, 48, 72, 100, 121, 108, 169, 196, 225, 192, 289, 288, 361, 300, 441, 484, 529, 432, 600, 676, 648, 588, 841, 900, 961, 768, 1089, 1156, 1225, 864, 1369, 1444, 1521, 1200, 1681, 1764, 1849, 1452, 1800, 2116, 2209, 1728, 2352, 2400

Offset: 1

Author

Álvar Ibeas, Jan 31 2015

Keywords

• mult
• nonn

Comments

The Dirichlet convolution of a(n) and sigma(n) is sigma(n^2).

Links

• Álvar Ibeas, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A000203, A034676, A063659, A065764.

Programs

PARI

a(n) = n^2*sumdiv(n, d, if (issquare(d), moebius(sqrtint(d))/d)); \\ Michel Marcus, Feb 10 2015

Formula

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

Multiplicative with a(p) = p^2; a(p^e) = p^(2e) - p^(2e-2), for e > 1.

Dirichlet g.f.: zeta(s-2) / zeta(2s-2).

Sum_{k=1..n} a(k) ~ 30 * n^3 / Pi^4. - Vaclav Kotesovec, Jan 11 2019

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/p^2 + 1/(p^2 - 1)^2) = 1.681923034881403168503816690236967736500606659628336043348190538886262268... - Vaclav Kotesovec, Sep 20 2020

a(n) = n*A063659(n). - Ridouane Oudra, Jul 26 2025