A052410 Write n = m^k with m, k integers, k >= 1, then a(n) is the smallest possible choice for m.

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 5, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 2, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

Value of m in m^p = n, where p is the largest possible power (see A052409).

For n > 1, n is a perfect power iff a(n) <> n. - Reinhard Zumkeller, Oct 13 2002

a(n)^A052409(n) = n. - Reinhard Zumkeller, Apr 06 2014

Every integer root of n is a power of a(n). All entries (except 1) belong to A007916. - Gus Wiseman, Sep 11 2017

Links

• Daniel Forgues, Table of n, a(n) for n=1..100000
• Eric Weisstein's World of Mathematics, Power
• Eric Weisstein's World of Mathematics, Perfect Power

Crossrefs

Cf. A001597, A025478, A007916, A027748, A052409, A072775, A124010, A175781, A278028, A288636, A289023.

Programs

Haskell

a052410 n = product $ zipWith (^)
                      (a027748_row n) (map (`div` (foldl1 gcd es)) es)
            where es = a124010_row n
-- Reinhard Zumkeller, Jul 15 2012

Maple

a:= n-> (l-> (t-> mul(i[1]^(i[2]/t), i=l))(
         igcd(seq(i[2], i=l))))(ifactors(n)[2]):
seq(a(n), n=1..74);  # Alois P. Heinz, Jul 22 2024

Mathematica

Table[If[n==1, 1, n^(1/(GCD@@(Last/@FactorInteger[n])))], {n, 100}]

PARI

a(n) = if (ispower(n,,&r), r, n); \\ Michel Marcus, Jul 19 2017

Python

def upto(n):
    list = [1] + [0] * (n - 1)
    for i in range(2, n + 1):
        if not list[i - 1]:
            j = i
            while j <= n:
                list[j - 1] = i
                j *= i
    return list
# M. Eren Kesim, Jun 03 2021

Python

from math import gcd
from sympy import integer_nthroot, factorint
def A052410(n): return integer_nthroot(n,gcd(*factorint(n).values()))[0] if n>1 else 1 # Chai Wah Wu, Mar 02 2024

Formula

a(A001597(k)) = A025478(k).

a(n) = A007916(A278028(n,1)). - Gus Wiseman, Sep 11 2017

Extensions

Definition edited (in a complementary form to A052409) by Daniel Forgues, Mar 14 2009

Corrected by Charles R Greathouse IV, Sep 02 2009

Definition edited by N. J. A. Sloane, Sep 03 2010