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

1,4

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

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

Formula

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

Example

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.

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 *)

Prog

(Haskell)
a052409 1 = 0
a052409 n = foldr1 gcd $ a124010_row n
-- Reinhard Zumkeller, May 26 2012
(PARI) a(n)=my(k=ispower(n)); if(k, k, n>1) \\ Charles R Greathouse IV, Oct 30 2012
(Scheme) (define (A052409 n) (if (= 1 n) 0 (gcd (A067029 n) (A052409 (A028234 n))))) ;; Antti Karttunen, Aug 07 2017
(Python)
from math import gcd
from sympy import factorint
def A052409(n): return gcd(*factorint(n).values()) # Chai Wah Wu, Aug 31 2022

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.

Sequence in context: A158052 A253641 A158378 * A327503 A051904 A070012

Adjacent sequences: A052406 A052407 A052408 * A052410 A052411 A052412

Keyword

nonn

Author

Eric W. Weisstein

Extensions

More terms from Labos Elemer, Jun 17 2002

Status

approved