A112526 Characteristic function for powerful numbers.

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

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

Links

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

Eric Weisstein's World of Mathematics, Powerful Number.

Index entries for characteristic functions.

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) * A008966(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011

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

a(n) = Sum_{d|n} A005361(d)*A008683(n/d). - Ridouane Oudra, Jul 03 2025

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
(PARI) a(n) = ispowerful(n); \\ Amiram Eldar, Jul 02 2025
(Python)
from sympy import factorint
def A112526(n): return int(all(e>1 for e in factorint(n).values())) # Chai Wah Wu, Sep 15 2024

Crossrefs

Differs from characteristic function of perfect powers A075802 at Achilles numbers A052486.

Cf. A001694 (powerful numbers), A124010, A001221, A027746.

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

Cf. A002117, A078434, A005361.

Sequence in context: A355448 A354868 A075802 * A307423 A355684 A355683

Adjacent sequences: A112523 A112524 A112525 * A112527 A112528 A112529

Keyword

mult,nonn,easy

Author

Franklin T. Adams-Watters, Sep 09 2005

Status

approved