A164572 Numbers k such that k and k+4 are both prime powers.

1, 3, 4, 5, 7, 9, 13, 19, 23, 25, 27, 37, 43, 49, 67, 79, 97, 103, 109, 121, 127, 163, 169, 193, 223, 229, 239, 277, 289, 307, 313, 343, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 729, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087

Offset: 1

Author

Daniel Forgues, Aug 16 2009, Aug 17 2009

Keywords

• nonn

Comments

Numbers n such that n + (0, 4) is a prime power pair.

A generalization of the cousin primes. The cousin primes are a subsequence.

n + (0, 2m), m >= 1, being an admissible pattern for prime pairs, since (0, 2m) = (0, 0) (mod 2), has high density.

n + (0, 2m-1), m >= 1, being a non-admissible pattern for prime pairs, since (0, 2m-1) = (0, 1) (mod 2), has low density [the only possible pairs are (2^a - 2m-1, 2^a) or (2^a, 2^a + 2m-1), a >= 0.]

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1256 from Daniel Forgues)

Crossrefs

Cf. A023200, A000961.

k and (x) are prime powers: A006549 (k+1), A120431 (k+2), A164571 (k+3), this sequence (k+4), A164573 (k+5), A164574 (k+6).

Programs

Mathematica

Select[Range[1000], PrimeNu[#] < 2 && PrimeNu[# + 4] < 2 &] (* Amiram Eldar, Oct 01 2020 *)

PARI

is(n)=if(n==1,return(1)); isprimepower(n) && isprimepower(n+4) \\ Charles R Greathouse IV, Apr 24 2015