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A243405 Minimum among the numbers p^(n/p), where p is a prime factor of n.

%I A243405 #8 Jun 16 2014 01:40:41
%S A243405 1,2,3,4,5,8,7,16,27,25,11,64,13,49,125,256,17,512,19,625,343,121,23,
%T A243405 4096,3125,169,19683,2401,29,15625,31,65536,1331,289,16807,262144,37,
%U A243405 361,2197,390625,41,117649,43,14641,1953125,529,47,16777216,823543,9765625,4913,28561,53
%N A243405 Minimum among the numbers p^(n/p), where p is a prime factor of n.
%C A243405 The setting a(1)=1 is conventional.
%C A243405 Upper bound (for any n): a(n) <= (3^(1/3))^n = A002581^n.
%H A243405 Stanislav Sykora, <a href="/A243405/b243405.txt">Table of n, a(n) for n = 1..2000</a>
%F A243405 For prime p, a(p)=p.
%F A243405 For n>1: When gpf(n)>3 then a(n)=gpf(n)^(n/gpf(n)); otherwise if n is even then a(n)=2^(n/2); otherwise a(n)=3^(n/3).
%F A243405 If n is in A033845, a(n) = 2^(n/2); otherwise a(n) = gpf(n)^(n/gpf(n)). - _Franklin T. Adams-Watters_, Jun 15 2014
%e A243405 a(12)=64 because 2^(12/2)=64 is smaller than 3^(12/3)=81.
%o A243405 (PARI) A243405(n)= {my(m,k,p,q);if(n==1,return(1));
%o A243405   p=factor(n);m=2^n;
%o A243405   for(k=1,#p[,1],q=p[k,1]^(n\p[k,1]);if(q<m,m=q));
%o A243405   return (m);}
%Y A243405 Cf. A002581, A092975 (maximum instead of minimum), A033845.
%K A243405 nonn
%O A243405 1,2
%A A243405 _Stanislav Sykora_, Jun 04 2014