A052409 - OEIS

%I #58 Aug 31 2022 10:51:21

%S 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,

%T 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,

%U 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

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

%C 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

%C 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

%C A052410(n)^a(n) = n. - _Reinhard Zumkeller_, Apr 06 2014

%C Positions of 1's are A007916. Smallest base is given by A052410. - _Gus Wiseman_, Jun 09 2020

%H Daniel Forgues, <a href="/A052409/b052409.txt">Table of n, a(n) for n = 1..100000</a>

%H N. J. A. Sloane, <a href="/A278028/a278028.txt">Maple programs for A007916, A278028, A278029, A052409, A089723, A277564</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Power.html">Power</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectPower.html">Perfect Power</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(1) = 0; for n > 1, a(n) = gcd(A067029(n), a(A028234(n))). - _Antti Karttunen_, Aug 07 2017

%e 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.

%e For n = 10800 = 2^4 * 3^3 * 5^2, GCD[4,3,2] = 1, thus a(10800) = 1.

%p # See link.

%p #

%p a:= n-> igcd(map(i-> i[2], ifactors(n)[2])[]):

%p seq(a(n), n=1..120);  # _Alois P. Heinz_, Oct 20 2019

%t Table[GCD @@ Last /@ FactorInteger[n], {n, 100}] (* _Ray Chandler_, Jan 24 2006 *)

%o (Haskell)

%o a052409 1 = 0

%o a052409 n = foldr1 gcd $ a124010_row n

%o -- _Reinhard Zumkeller_, May 26 2012

%o (PARI) a(n)=my(k=ispower(n)); if(k, k, n>1) \\ _Charles R Greathouse IV_, Oct 30 2012

%o (Scheme) (define (A052409 n) (if (= 1 n) 0 (gcd (A067029 n) (A052409 (A028234 n))))) ;; _Antti Karttunen_, Aug 07 2017

%o (Python)

%o from math import gcd

%o from sympy import factorint

%o def A052409(n): return gcd(*factorint(n).values()) # _Chai Wah Wu_, Aug 31 2022

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

%Y Apart from the initial term essentially the same as A253641.

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

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

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

%K nonn

%O 1,4

%A _Eric W. Weisstein_

%E More terms from _Labos Elemer_, Jun 17 2002