A075802 Characteristic function of perfect powers, A001597.

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, 0, 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

Reinhard Zumkeller, Oct 13 2002

Keywords

• nonn

Comments

Not multiplicative: for example, a(8)=a(9)=1, but a(72)=0. - Franklin T. Adams-Watters, Sep 09 2005

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Perfect Powers.
• Index entries for characteristic functions

Crossrefs

Cf. A001597, A007916, A052409, A057427, A072292, A112526.

Programs

Haskell

a075802 1 = 1
a075802 n = signum $ a052409 n - 1  -- Reinhard Zumkeller, May 26 2012

Mathematica

a[n_] := Boole[GCD @@ FactorInteger[n][[All, 2]] > 1]; a[1] = 1; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Dec 12 2011 *)

Python

from sympy import perfect_power
def A075802(n): return int(bool(perfect_power(n))) if n>1 else 1 # Chai Wah Wu, Mar 11 2025

Formula

a(n) = A057427(A052409(n) - 1);

a(A001597(n))=1 and a(A007916(n))=0.