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A112526
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Characteristic function for powerful numbers.
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23
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1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0, 0, 1, 0, 0, 0, 0, 0
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OFFSET
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1,1
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COMMENTS
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A signed multiplicative variant is defined by b(n) = a(n)*mu(n) with mu = A008683, such that b(p^e)=0 if e=1 and b(p^e)= -1 if e>1. This has Dirichlet series sum_{n>=1} b(n)/n = A005596 and sum_{n>=1} b(n)/n^2 = A065471. - R. J. Mathar, Apr 04 2011
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LINKS
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FORMULA
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Multiplicative with a(p^e) = 1 - 0^(e-1), e>0 and p prime.
Dirichlet g.f.: zeta(2*s)*zeta(3*s)/zeta(6*s), e.g. A082695 at s=1.
Sum_{k=1..n} a(k) ~ Zeta(3/2)*sqrt(n)/Zeta(3) + 6*Zeta(2/3)*n^(1/3)/Pi^2. - Vaclav Kotesovec, Feb 08 2019
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EXAMPLE
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a(72)=1 because 72=2^3*3^2 has all exponents > 1.
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MATHEMATICA
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cfpn[n_]:=If[n==1||Min[Transpose[FactorInteger[n]][[2]]]>1, 1, 0]; Array[ cfpn, 120] (* Harvey P. Dale, Jul 17 2012 *)
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PROG
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(Haskell)
a112526 1 = 1
a112526 n = fromEnum $ (> 1) $ minimum $ a124010_row n
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1+X^3)/(1-X^2))[n], ", ")) \\ Vaclav Kotesovec, Jul 15 2022
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CROSSREFS
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KEYWORD
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mult,nonn
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AUTHOR
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STATUS
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approved
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