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

1,3

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.

Links

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

Example

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.

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}]

Crossrefs

Cf. A000040, A001597, A276711.

Sequence in context: A279875 A185872 A186710 * A374777 A318491 A060640

Adjacent sequences: A276714 A276715 A276716 * A276718 A276719 A276720

Keyword

nonn

Author

Zhi-Wei Sun, Sep 16 2016

Status

approved