A276717 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.

1, 1, 5, 7, 17, 11, 13, 37, 17, 19, 89, 19, 41, 71, 29, 13, 73, 199, 37, 157, 41, 43, 17, 47, 113, 433, 53, 541, 809, 59, 61, 997, 89, 67, 1009, 71, 73, 113, 521, 79, 1553, 83, 1721, 1693, 89, 1873, 1697, 107, 97, 313, 101, 103, 761, 107, 109, 11, 113, 239, 1433, 2269

Offset: 1

Author

Zhi-Wei Sun, Sep 16 2016

Keywords

• nonn

Comments

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

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

Examples

			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.
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.
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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000040, A001597, A276711.

Programs

Mathematica

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}]