A270492 -id:A270492 - cp's OEIS frontend

A290084 a(1) = a(2) = 1; for n > 2, a(n) = A000010(n) / A270492(n).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 1, 4, 6, 1, 1, 4, 1, 1, 1, 6, 1, 4, 1, 8, 10, 1, 12, 6, 1, 1, 12, 8, 1, 6, 1, 10, 12, 1, 1, 8, 1, 1, 16, 12, 1, 1, 20, 12, 18, 1, 1, 8, 1, 1, 6, 16, 12, 10, 1, 16, 22, 12, 1, 12, 1, 1, 20, 18, 30, 12, 1, 16, 1, 1, 1, 12, 16, 1, 28, 20, 1, 12, 12, 22, 30, 1, 36, 16, 1, 1, 30, 20, 1, 16, 1, 24, 24

Offset: 1

Antti Karttunen, Aug 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A034380, A270492.

(define (A290084 n) (if (<= n 2) 1 (/ (A000010 n) (A270492 n))))

a(1) = a(2) = 1; for n > 2, a(n) = A000010(n) / A270492(n).

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

Original entry on oeis.org

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

Eric W. Weisstein

nonn

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

			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.

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

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.

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

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

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

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

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

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

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

More terms from Labos Elemer, Jun 17 2002

cp's OEIS Frontend

A290084 a(1) = a(2) = 1; for n > 2, a(n) = A000010(n) / A270492(n).

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

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