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, 2601

Offset

1,2

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

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

Mathematica

f[p_, e_] := If[e == 1, p^2, p^(2*e) - p^(2*e-2)]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 24 2025 *)

Prog

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

Crossrefs

Cf. A000290, A000203, A008683, A034676, A063659, A065764.

Sequence in context: A377280 A297414 A076794 * A344578 A283105 A390251

Adjacent sequences: A254517 A254518 A254519 * A254521 A254522 A254523

Keyword

mult,nonn,easy

Author

Álvar Ibeas, Jan 31 2015

Status

approved