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A112526 Characteristic function for powerful numbers. 23
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
a(n) * A008966(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011
LINKS
Eric Weisstein's World of Mathematics, Powerful Number.
FORMULA
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.
a(n) = {m: Min{A124010(m,k): k=1..A001221(m)} > 1}. - Reinhard Zumkeller, Jun 03 2015
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
EXAMPLE
a(72)=1 because 72=2^3*3^2 has all exponents > 1.
MATHEMATICA
cfpn[n_]:=If[n==1||Min[Transpose[FactorInteger[n]][[2]]]>1, 1, 0]; Array[ cfpn, 120] (* Harvey P. Dale, Jul 17 2012 *)
PROG
(Haskell)
a112526 1 = 1
a112526 n = fromEnum $ (> 1) $ minimum $ a124010_row n
-- Reinhard Zumkeller, Jun 03 2015, Sep 16 2011
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1+X^3)/(1-X^2))[n], ", ")) \\ Vaclav Kotesovec, Jul 15 2022
CROSSREFS
Powerful numbers: A001694. Differs from perfect powers A075802 at Achilles numbers A052486.
Sequence in context: A355448 A354868 A075802 * A307423 A355684 A355683
KEYWORD
mult,nonn
AUTHOR
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)