A052409 a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).

0, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

Greatest common divisor of all prime-exponents in canonical factorization of n for n>1: a(n)>1 iff n is a perfect power; a(A001597(k))=A025479(k). - Reinhard Zumkeller, Oct 13 2002

a(1) set to 0 since there is no largest finite integer power m for which a representation of the form 1 = 1^m exists (infinite largest m). - Daniel Forgues, Mar 06 2009

A052410(n)^a(n) = n. - Reinhard Zumkeller, Apr 06 2014

Positions of 1's are A007916. Smallest base is given by A052410. - Gus Wiseman, Jun 09 2020

Examples

			n = 72 = 2*2*2*3*3: GCD[exponents] = GCD[3,2] = 1. This is the least n for which a(n) <> A051904(n), the minimum of exponents.
For n = 10800 = 2^4 * 3^3 * 5^2, GCD[4,3,2] = 1, thus a(10800) = 1.
		

Links

• Daniel Forgues, Table of n, a(n) for n = 1..100000
• N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
• Eric Weisstein's World of Mathematics, Power
• Eric Weisstein's World of Mathematics, Perfect Power
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A052410, A005361, A051903, A072411-A072414, A124010, A075802, A072776, A270492.

Apart from the initial term essentially the same as A253641.

Differs from A051904 for the first time at n=72, where a(72) = 1, while A051904(72) = 2.

Differs from A158378 for the first time at n=10800, where a(10800) = 1, while A158378(10800) = 2.

Cf. A000005, A000961, A001597, A052410, A303386, A327501.

Programs

Haskell

a052409 1 = 0
a052409 n = foldr1 gcd $ a124010_row n
-- Reinhard Zumkeller, May 26 2012

Maple

# See link.
#
a:= n-> igcd(map(i-> i[2], ifactors(n)[2])[]):
seq(a(n), n=1..120);  # Alois P. Heinz, Oct 20 2019

Mathematica

Table[GCD @@ Last /@ FactorInteger[n], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

PARI

a(n)=my(k=ispower(n)); if(k, k, n>1) \\ Charles R Greathouse IV, Oct 30 2012

Python

from math import gcd
from sympy import factorint
def A052409(n): return gcd(*factorint(n).values()) # Chai Wah Wu, Aug 31 2022

Scheme

(define (A052409 n) (if (= 1 n) 0 (gcd (A067029 n) (A052409 (A028234 n))))) ;; Antti Karttunen, Aug 07 2017

Formula

a(1) = 0; for n > 1, a(n) = gcd(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 07 2017

Extensions

More terms from Labos Elemer, Jun 17 2002