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A112526 Characteristic function for powerful numbers. 8
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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for characteristic functions

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

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

CROSSREFS

Powerful numbers: A001694. Differs from perfect powers A075802 at Achilles numbers A052486.

Cf. A124010, A001221, A027746.

Cf. A008683, A008966, A005596, A065471, A082695, A063524.

Sequence in context: A205633 A170956 A075802 * A120523 A030315 A119498

Adjacent sequences:  A112523 A112524 A112525 * A112527 A112528 A112529

KEYWORD

mult,nonn

AUTHOR

Franklin T. Adams-Watters, Sep 09 2005

STATUS

approved

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Last modified August 20 03:34 EDT 2017. Contains 290823 sequences.