A063659 -id:A063659 - cp's OEIS frontend

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

Álvar Ibeas, Jan 31 2015

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

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

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

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

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

A301511 Expansion of e.g.f. exp(Sum_{k>=1} psi(k)*x^k/k!), where psi() is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 1, 4, 14, 68, 362, 2224, 14940, 110348, 878600, 7518002, 68529122, 662709832, 6764329158, 72622813172, 817239648500, 9612724174088, 117878757097178, 1503660164683864, 19911519090176808, 273221610513382028, 3878513600608651636, 56873187579428449852, 860296560100458300892

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Exponential transform of A001615.

			E.g.f.: A(x) = 1 + x/1! + 4*x^2/2! + 14*x^3/3! + 68*x^4/4! + 362*x^5/5! + 2224*x^6/6! + 14940*x^7/7! + ...

Eric Weisstein's World of Mathematics, Dedekind Function
N. J. A. Sloane, Transforms

Cf. A001615, A063659, A156303, A300011.

psi[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]; a[n_] := a[n] = SeriesCoefficient[Exp[Sum[psi[k] x^k/k!, {k, 1, n}]], {x, 0, n}]; Table[a[n] n!, {n, 0, 23}]
psi[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]; a[n_] := a[n] = Sum[psi[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

E.g.f.: exp(Sum_{k>=1} A001615(k)*x^k/k!).

cp's OEIS Frontend

A254520 Möbius transform of A034676.

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