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

%I A240981 #21 Aug 11 2014 04:43:08
%S A240981 25,1,4,1,16,36,8,8,4,196,1,27,8,441,81,676,841,64,1089,125,8,49,1681,
%T A240981 32,128,27,25,2197,16,8,1,125,32,2048,2048,361,243,6561,49,7396,64,
%U A240981 8100,25,32,6859,125,32,289,16,27,128,4,243,1936,32,17161,243,729
%N A240981 Smallest perfect power b^e such that b^e+prime(n) is also a perfect power.
%C A240981 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.
%C A240981 For all n, a(n) exists.
%C A240981 Subsequence of A103953.
%C A240981 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
%H A240981 Jens Kruse Andersen, <a href="/A240981/b240981.txt">Table of n, a(n) for n = 1..10000</a>
%e A240981 n=1: prime(n)=2, 25 = 5^2 and  25+5=27=3^3;
%e A240981 n=2: prime(n)=3, 1 = 1^2 and  1+3=4=2^2;
%e A240981 n=3: prime(n)=5, 4 = 2^2 and  4+5=9=3^2.
%o A240981 (PARI) {forprime(p=2,200,if(ispower(1+p),print1(1","),n=4;while(!(ispower(n)&&ispower(n+p)),n++);print1(n",")))}
%Y A240981 Cf. A000668, A001597, A103953.
%K A240981 nonn
%O A240981 1,1
%A A240981 _Zak Seidov_, Aug 06 2014