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A317531 Expansion of Sum_{p prime, k>=1} x^(p^k)/(1 + x^(p^k)). 2
0, 1, 1, 0, 1, 0, 1, -1, 2, 0, 1, -1, 1, 0, 2, -2, 1, -1, 1, -1, 2, 0, 1, -2, 2, 0, 3, -1, 1, -1, 1, -3, 2, 0, 2, -2, 1, 0, 2, -2, 1, -1, 1, -1, 3, 0, 1, -3, 2, -1, 2, -1, 1, -2, 2, -2, 2, 0, 1, -2, 1, 0, 3, -4, 2, -1, 1, -1, 2, -1, 1, -3, 1, 0, 3, -1, 2, -1, 1, -3, 4, 0, 1, -2, 2, 0, 2, -2, 1, -2, 2, -1, 2, 0, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

G.f.: Sum_{k>=1} x^A246655(k)/(1 + x^A246655(k)).

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

a(n) = Sum_{d|n} (-1)^(n/d+1)*A069513(d).

If n is odd, a(n) = A001222(n).

MATHEMATICA

nmax = 95; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k]] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + Boole[PrimePowerQ[k]] x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

Table[DivisorSum[n, (-1)^(n/# + 1) &, PrimePowerQ[#] &], {n, 95}]

PROG

(PARI) A317531(n) = sumdiv(n, d, ((-1)^(n/d+1))*(1==omega(d))); \\ Antti Karttunen, Sep 30 2018

CROSSREFS

Cf. A001222, A048272, A069513, A246655, A288571, A305614, A317528.

Sequence in context: A333781 A243223 A034178 * A074169 A328458 A099362

Adjacent sequences:  A317528 A317529 A317530 * A317532 A317533 A317534

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Jul 30 2018

STATUS

approved

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Last modified May 31 22:45 EDT 2020. Contains 334756 sequences. (Running on oeis4.)