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

1,2

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

Formula

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

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

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}]

Prog

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

Crossrefs

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

Sequence in context: A382904 A243074 A304776 * A327501 A175781 A072775

Adjacent sequences: A052407 A052408 A052409 * A052411 A052412 A052413

Keyword

nonn

Author

Eric W. Weisstein

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

Status

approved