A276717 - OEIS

%I #14 Sep 18 2016 20:16:34

%S 1,1,5,7,17,11,13,37,17,19,89,19,41,71,29,13,73,199,37,157,41,43,17,

%T 47,113,433,53,541,809,59,61,997,89,67,1009,71,73,113,521,79,1553,83,

%U 1721,1693,89,1873,1697,107,97,313,101,103,761,107,109,11,113,239,1433,2269

%N Least prime p < n^2 such that n^2 - p = x^k for some integers x > 1 and k > 1, or 1 if such a prime p does not exist.

%C The conjecture in A276711 implies that a(n) > 1 for all n > 2 except for n = 11^3 = 1331.

%C Note that for any integer n > 2 neither n^2 nor n^2 - 1 could be a prime.

%H Zhi-Wei Sun, <a href="/A276717/b276717.txt">Table of n, a(n) for n = 1..5000</a>

%e a(2) = 1 since neither 2^2 - 2 nor 2^2 -3 has the form x^k with x and k integers greater than one.

%e a(3) = 5 since 5 is a prime with 3^2 - 5 = 2^2 but neither 3^2 - 2 nor 3^2 - 3 is a perfect power.

%e a(4913) = 23613281 since 23613281 is a prime with 4913^2 - 23613281 = 2^19, and 4913^2 - p is not a perfect power for any prime p < 23613281.

%t Do[Do[If[IntegerQ[(n^2-Prime[j])^(1/k)],Print[n," ",Prime[j]];Goto[aa]],{j,1,PrimePi[n^2-2]},{k,2,Log[2,n^2-Prime[j]]}];Print[n," ",1];Label[aa];Continue,{n,1,60}]

%Y Cf. A000040, A001597, A276711.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Sep 16 2016