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A326813 Dirichlet g.f.: zeta(2*s) / (1 - 2^(-s)). 1
1, 1, 0, 2, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Table of n, a(n) for n=1..100.

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

FORMULA

G.f.: Sum_{k>=0} (theta_3(x^(2^k)) - 1) / 2.

a(n) = Sum_{d|n} A209229(n/d) * A010052(d).

a(n) = Sum_{d|n} tau(n/d) * (-1)^bigomega(d) * 2^(omega(2*d) - 1), where tau = A000005, bigomega = A001222 and omega = A001221.

Product_{n>=1} (1 + x^n)^a(n) = g.f. for A001156.

Sum_{k=1..n} a(k) ~ sqrt(2*n) / (sqrt(2)-1). - Vaclav Kotesovec, Oct 20 2019

MATHEMATICA

Table[Sum[Boole[IntegerQ[Log[2, n/d]]] Boole[IntegerQ[d^(1/2)]], {d, Divisors[n]}], {n, 1, 100}]

Table[Sum[DivisorSigma[0, n/d] (-1)^PrimeOmega[d] 2^(PrimeNu[2 d] - 1), {d, Divisors[n]}], {n, 1, 100}]

CROSSREFS

Cf. A000005, A001156, A001221, A001222, A001511, A010052, A028983 (positions of 0's), A053866, A209229.

Sequence in context: A088886 A317636 A305566 * A137347 A024941 A219492

Adjacent sequences:  A326810 A326811 A326812 * A326814 A326815 A326816

KEYWORD

nonn,mult

AUTHOR

Ilya Gutkovskiy, Oct 19 2019

STATUS

approved

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Last modified September 27 11:41 EDT 2020. Contains 337380 sequences. (Running on oeis4.)