A120431 Numbers k such that k and k+2 are prime powers.

1, 2, 3, 5, 7, 9, 11, 17, 23, 25, 27, 29, 41, 47, 59, 71, 79, 81, 101, 107, 125, 137, 149, 167, 179, 191, 197, 227, 239, 241, 269, 281, 311, 347, 359, 419, 431, 461, 521, 569, 599, 617, 641, 659, 727, 809, 821, 827, 839, 857, 881, 1019, 1031, 1049, 1061, 1091

Offset: 1

Author

Greg Huber, Jul 13 2006

Keywords

• nonn

Comments

Twin prime powers, a generalization of the twin primes. The twin primes are a subsequence.

From Daniel Forgues, Aug 17 2009: (Start)

Numbers k such that k + (0, 2) is a prime power pair.

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

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

Examples

			a(5) = 7 since the 5th pair of twin prime powers is (7,9), while the first four pairs are (1,3), (2,4), (3,5) and (5,7).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1270 from Daniel Forgues)

Crossrefs

Cf. A001359, A000961, A064076.

Cf. A006549, A164571, A164572, A164573, A164574.

Programs

Magma

[1] cat [n: n in [2..1200] | IsPrimePower(n) and IsPrimePower(n+2)]; // Vincenzo Librandi, Nov 03 2018

Maple

isppow := proc(n) local pf; pf := ifactors(n)[2]; if nops(pf) = 1 or n =1 then true; else false; fi; end; isA120431 := proc(n) RETURN (isppow(n) and isppow(n+2)); end; for n from 1 to 1500 do if isA120431(n) then printf("%d, ",n); fi; od; # R. J. Mathar, Dec 16 2006

Mathematica

Join[{1}, Select[Range[1100], And@@PrimePowerQ/@{#, # + 2} &]] (* Vincenzo Librandi, Nov 03 2018 *)

PARI

is(n)=if(n<4,return(n>0)); isprimepower(n) && isprimepower(n+2) \\ Charles R Greathouse IV, Apr 24 2015

Formula

a(n) = A064076(n-2) for n >= 3. - Georg Fischer, Nov 02 2018

Extensions

More terms from R. J. Mathar, Dec 16 2006