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

1,2

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)

Links

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

Formula

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

Example

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).

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 *)

Prog

(PARI) is(n)=if(n<4, return(n>0)); isprimepower(n) && isprimepower(n+2) \\ Charles R Greathouse IV, Apr 24 2015
(Magma) [1] cat [n: n in [2..1200] | IsPrimePower(n) and IsPrimePower(n+2)]; // Vincenzo Librandi, Nov 03 2018

Crossrefs

Cf. A001359, A000961, A064076.

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

Sequence in context: A281541 A066824 A036963 * A214548 A254440 A284914

Adjacent sequences: A120428 A120429 A120430 * A120432 A120433 A120434

Keyword

nonn

Author

Greg Huber, Jul 13 2006

Extensions

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

Status

approved