A267945 Primes that are a prime power plus two.

5, 7, 11, 13, 19, 29, 31, 43, 61, 73, 83, 103, 109, 127, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883

Offset: 1

Author

Robert C. Lyons, Jan 22 2016

Keywords

• nonn

Comments

The term 'prime power' refers to the elements of A246655.

If we were to extend the definition of prime power to include 1, then 3 would be the first term of the sequence, because 3 = 2^0 + 2.

The sequence is probably infinite, since it includes all the terms of A006512 (Greater of twin primes).

From Robert Israel, Jan 22 2016: (Start)

Since 3 divides p or p^k+2 if k is even, the only terms of the form p^k+2 where k is even are A228034.

All terms not in A057735 are congruent to 1 mod 3.

The generalized Bunyakovsky conjecture implies that for any odd k, there are infinitely many terms of the form p^k+2. (End)

Examples

			5 is in the sequence because 5 = 3^1 + 2.
7 is in the sequence because 7 = 5^1 + 2.
11 is in the sequence because 11 = 3^2 + 2.
13 is in the sequence because 13 = 11^1 + 2.
29 is in the sequence because 29 = 3^3 + 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Wikipedia, Generalized Bunyakovsky conjecture

Crossrefs

Cf. A000961, A057735, A228034, A246655, A267944.

Programs

Maple

select(t -> isprime(t) and nops(numtheory:-factorset(t-2))=1, [ seq(i,i=3..1000, 2)]); # Robert Israel, Jan 22 2016

Mathematica

A267945Q = PrimeQ@# && (Length@# == 1 && #[[1, 1]] > 1 &@FactorInteger[# - 2]) & (* JungHwan Min, Jan 25 2016 *)
Select[Array[Prime, 100], Length@# == 1 && #[[1, 1]] > 1 &@FactorInteger[# - 2] &] (* JungHwan Min, Jan 25 2016 *)

PARI

lista(nn) = {forprime(p=2, nn, if (isprimepower(p-2), print1(p, ", ")););} \\ Michel Marcus, Jan 22 2016

Sage

filter( is_prime, [ n+2 for n in prime_powers( 1, 1000 ) ] )