A267944 -id:A267944 - cp's OEIS frontend

A053705 Primes p of form q^k-2 where q is also a prime and k > 1.

2, 7, 23, 47, 79, 167, 241, 359, 727, 839, 1367, 1847, 2207, 2399, 3719, 5039, 6857, 7919, 10607, 11447, 14639, 16127, 17159, 19319, 19681, 28559, 29789, 29927, 36479, 44519, 49727, 50651, 54287, 57119, 66047, 85847, 97967, 113567, 128879

Offset: 1

Labos Elemer, Feb 14 2000

nonn

			79 = 3^4-2.
241 = 3^5-2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A025475, A053704.

Subsequence of A267944.

Do[s=2+Prime[n]; If[Equal[Length[FactorInteger[s]], 1]&&!PrimeQ[s], Print[s-2]], {n, 1, 100000}]
fQ[n_] := PrimeNu[n + 2] == 1 && ! PrimeQ[n + 2]; Select[ Prime@ Range@ 15000, fQ] (* Robert G. Wilson v, Apr 01 2012 *)
seq[max_] := Module[{s = {}, p = 2}, While[p^2 <= max, s = Join[s, Select[p^Range[2, Floor[Log[p, max]]], PrimeQ[# - 2] &]]; p = NextPrime[p]]; Union[s] - 2]; seq[150000] (* Amiram Eldar, Aug 27 2024 *)

lista(nn) = forprime (p=1, nn, if (ispower(p+2,,&q) && isprime(q), print1(p, ", ")); ); \\ Michel Marcus, Dec 11 2014

a(n) = A053704(n) - 2. - Amiram Eldar, Aug 27 2024

Definition corrected by Zak Seidov, Dec 11 2014

A267945 Primes that are a prime power plus two.

Original entry on oeis.org

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

Robert C. Lyons, Jan 22 2016

nonn

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)

			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.

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

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

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

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 *)

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

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

cp's OEIS Frontend

A053705 Primes p of form q^k-2 where q is also a prime and k > 1.

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