A240981 Smallest perfect power b^e such that b^e+prime(n) is also a perfect power.

25, 1, 4, 1, 16, 36, 8, 8, 4, 196, 1, 27, 8, 441, 81, 676, 841, 64, 1089, 125, 8, 49, 1681, 32, 128, 27, 25, 2197, 16, 8, 1, 125, 32, 2048, 2048, 361, 243, 6561, 49, 7396, 64, 8100, 25, 32, 6859, 125, 32, 289, 16, 27, 128, 4, 243, 1936, 32, 17161, 243, 729

Offset: 1

Author

Zak Seidov, Aug 06 2014

Keywords

• nonn

Comments

a(n) = 1 if and only if prime(n) is a Mersenne prime (A000668). Thus A059305 gives the values of n for which a(n) = 1.

For all n, a(n) exists.

Subsequence of A103953.

For n > 1, a(n) <= ((prime(n)-1)/2)^2, since ((p-1)/2)^2 + p = ((p+1)/2)^2. - Jens Kruse Andersen, Aug 10 2014

Examples

			n=1: prime(n)=2, 25 = 5^2 and  25+5=27=3^3;
n=2: prime(n)=3, 1 = 1^2 and  1+3=4=2^2;
n=3: prime(n)=5, 4 = 2^2 and  4+5=9=3^2.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000668, A001597, A103953.

Programs

PARI

{forprime(p=2,200,if(ispower(1+p),print1(1","),n=4;while(!(ispower(n)&&ispower(n+p)),n++);print1(n",")))}