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

1,1

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)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Wikipedia, Generalized Bunyakovsky conjecture

Example

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.

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 *)

Prog

(Sage) filter( is_prime, [ n+2 for n in prime_powers( 1, 1000 ) ] )
(PARI) lista(nn) = {forprime(p=2, nn, if (isprimepower(p-2), print1(p, ", ")); ); } \\ Michel Marcus, Jan 22 2016

Crossrefs

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

Sequence in context: A081759 A155053 A152468 * A088664 A023219 A343448

Adjacent sequences: A267942 A267943 A267944 * A267946 A267947 A267948

Keyword

nonn

Author

Robert C. Lyons, Jan 22 2016

Status

approved