A253641 Largest integer b such that n=a^b for some integer a; a(0)=a(1)=1 by convention.

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Author

M. F. Hasler, Jan 25 2015

Keywords

• nonn
• easy

Comments

A000005(a(n))-1 yields the number of times n is listed in A072103, i.e., the number of ways it can be written differently as perfect power.

For any n, a(n) >= 1 since n = n^1.

Integers n = 0 and n = 1 can be written as n^b with arbitrarily large b; to remain consistent with the preceding formula and the comment, the conventional choice a(n) = 1 seemed the best one.

The same as A052409 if the convention is dropped. - R. J. Mathar, Jan 29 2015

Examples

			a(4) = 2 since 4 = 2^2. a(64) = 6 since 64 = 2^6 (although also 64 = 4^3 = 8^2).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001597, A072103, A175064, A253642, A052409.

Programs

Maple

a:=proc(n) if n in {0, 1} then 1 else igcd(map(i->i[2], ifactors(n)[2])[]); fi; end: seq(a(n), n=0..120); # Ridouane Oudra, Jun 10 2025

PARI

A253641(n)=max(ispower(n),1)

Python

from math import gcd
from sympy import factorint
def A253641(n): return gcd(*factorint(n).values()) if n>1 else 1 # Chai Wah Wu, Aug 13 2024