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

Author

Franklin T. Adams-Watters, Sep 09 2005

Keywords

• mult
• nonn
• easy

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

Examples

			a(72) = 1 because 72 = 2^3*3^2 has all exponents > 1.
		

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.

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.

Programs

Haskell

a112526 1 = 1
a112526 n = fromEnum $ (> 1) $ minimum $ a124010_row n
-- Reinhard Zumkeller, Jun 03 2015, Sep 16 2011

Mathematica

cfpn[n_]:=If[n==1||Min[Transpose[FactorInteger[n]][[2]]]>1,1,0]; Array[ cfpn,120] (* Harvey P. Dale, Jul 17 2012 *)

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

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