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

1, 2, 3, 4, 5, 8, 7, 16, 27, 25, 11, 64, 13, 49, 125, 256, 17, 512, 19, 625, 343, 121, 23, 4096, 3125, 169, 19683, 2401, 29, 15625, 31, 65536, 1331, 289, 16807, 262144, 37, 361, 2197, 390625, 41, 117649, 43, 14641, 1953125, 529, 47, 16777216, 823543, 9765625, 4913, 28561, 53

Offset: 1

Author

Stanislav Sykora, Jun 04 2014

Keywords

• nonn

Comments

The setting a(1)=1 is conventional.

Upper bound (for any n): a(n) <= (3^(1/3))^n = A002581^n.

Examples

			a(12)=64 because 2^(12/2)=64 is smaller than 3^(12/3)=81.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A002581, A092975 (maximum instead of minimum), A033845.

Programs

PARI

A243405(n)= {my(m,k,p,q);if(n==1,return(1));
  p=factor(n);m=2^n;
  for(k=1,#p[,1],q=p[k,1]^(n\p[k,1]);if(q
				

Formula

For prime p, a(p)=p.

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).

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