A300894 -id:A300894 - cp's OEIS frontend

A300852 L.g.f.: log(Product_{k>=1} (1 + x^prime(k))) = Sum_{n>=1} a(n)*x^n/n.

0, 2, 3, -2, 5, -1, 7, -2, 3, -3, 11, -5, 13, -5, 8, -2, 17, -1, 19, -7, 10, -9, 23, -5, 5, -11, 3, -9, 29, -6, 31, -2, 14, -15, 12, -5, 37, -17, 16, -7, 41, -8, 43, -13, 8, -21, 47, -5, 7, -3, 20, -15, 53, -1, 16, -9, 22, -27, 59, -10, 61, -29, 10, -2, 18, -12, 67, -19, 26, -10, 71, -5, 73, -35, 8

Offset: 1

Ilya Gutkovskiy, Mar 13 2018

			L.g.f.: L(x) = 2*x^2/2 + 3*x^3/3 - 2*x^4/4 + 5*x^5/5 - x^6/6 + 7*x^7/7 - 2*x^8/8 + 3*x^9/9 - 3*x^10/10 + ...
exp(L(x)) = 1 + x^2 + x^3 + 2*x^5 + 2*x^7 + x^8 + x^9 + 2*x^10 + ... + A000586(n)*x^n + ...

Cf. A000040 (fixed points), A000586, A008472, A300893, A300894.

nmax = 75; Rest[CoefficientList[Series[Log[Product[(1 + x^Prime[k]), {k, 1, nmax}]], {x, 0, nmax}],x] Range[0, nmax]]
nmax = 75; Rest[CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 + x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, PrimeQ[#] &], {n, 75}]

G.f.: Sum_{k>=1} prime(k)*x^prime(k)/(1 + x^prime(k)).

a(n) = Sum_{p|n, p prime} p * (-1)^(n/p + 1). [See Mmca prog.]

A317528 Expansion of Sum_{k>=1} mu(k)^2*x^k/(1 + x^k), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 0, 2, -2, 2, 0, 2, -2, 2, 0, 2, -4, 2, 0, 4, -2, 2, 0, 2, -4, 4, 0, 2, -4, 2, 0, 2, -4, 2, 0, 2, -2, 4, 0, 4, -4, 2, 0, 4, -4, 2, 0, 2, -4, 4, 0, 2, -4, 2, 0, 4, -4, 2, 0, 4, -4, 4, 0, 2, -8, 2, 0, 4, -2, 4, 0, 2, -4, 4, 0, 2, -4, 2, 0, 4, -4, 4, 0, 2, -4, 2, 0, 2, -8, 4, 0, 4, -4, 2, 0, 4, -4, 4, 0, 4

Offset: 1

Ilya Gutkovskiy, Jul 30 2018

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A005117, A008683, A008966, A034444, A048272, A100007, A300894, A305614.

with(numtheory): seq(coeff(series(add(mobius(k)^2*x^k/(1+x^k),k=1..n), x,n+1),x,n),n=1..120); # Muniru A Asiru, Jul 30 2018

nmax = 95; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + MoebiusMu[k]^2 x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
Table[DivisorSum[n, (-1)^(n/# + 1) &, SquareFreeQ[#] &], {n, 95}]
f[p_, e_] := 2; f[2, e_] := If[e == 1, 0, -2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 19 2022 *)

A317528(n) = sumdiv(n,d,((-1)^(1+d))*issquarefree(n/d)); \\ Antti Karttunen, Dec 05 2021

G.f.: Sum_{k>=1} x^A005117(k)/(1 + x^A005117(k)).
L.g.f.: log(Product_{k>=1} (1 + mu(k)^2*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(n/d+1)*A008966(d).
If n is odd, a(n) = A034444(n).
Multiplicative with a(2) = 0, a(2^e) = -2 for e>1, and a(p^e) = 2 for p>2 and e>=1. - Amiram Eldar, Nov 19 2022

cp's OEIS Frontend

A300852 L.g.f.: log(Product_{k>=1} (1 + x^prime(k))) = Sum_{n>=1} a(n)*x^n/n.

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