A051779 Primes of form p*q + 2 where p and q are twin primes.

17, 37, 22501, 32401, 57601, 72901, 176401, 324901, 1664101, 1742401, 5336101, 6502501, 7452901, 11289601, 11492101, 18147601, 21622501, 34222501, 34574401, 40449601, 45968401, 81000001, 85377601, 92736901, 110880901, 118592101, 143280901, 187142401, 191268901

Offset

1,1

Comments

Starting with a(3) = 22501, all terms are of the form 900*n^2 + 1 with n=5, 6, 8, 9, 14, 19, 43, 44, 77, 85,... (A125251). - Zak Seidov, Dec 07 2008

Primes of the form (p^2 + q^2)/2, where p and q are twin primes. - Thomas Ordowski and Altug Alkan, Mar 19 2017

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Formula

From Jonathan Vos Post, May 11 2006: (Start)

{A037074(k) + 2} INTERSECT {A000040}.

{A001359(k) * A006512(k) + 2} INTERSECT {A000040}.

{A054735(k)^2 + 2*A054735(k) + 2} INTERSECT {A000040}. (End)

Example

22501 is a term because 22501 = 149*151 + 2, in which 22501 is prime and {149,151} is a twin prime pair.

Maple

with (numtheory): for n from 1 to 2000 do if (ithprime(n+1)-ithprime(n)=2) then if (tau(ithprime(n)*ithprime(n+1)+2)=2) then print(ithprime(n), ithprime(n+1), ithprime(n)*ithprime(n+1)+2); fi; fi; od;

Mathematica

lst={}; Do[p=Prime[n]; If[Length[Divisors[p-2]]==4&&(Divisors[p-2][[3]]-Divisors[p-2][[2]])==2, AppendTo[lst, p]], {n, 6*10^5}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 08 2008 *)
(* Alternative: *)
Select[(First[#]Last[#]+2)&/@Select[Partition[Prime[Range[2700]], 2, 1], Last[#]-First[#]==2&], PrimeQ]  (* Harvey P. Dale, Mar 11 2011 *)
(* Alternative: *)
Select[2 + Times @@@ Select[ Partition[ Prime@ Range@ 1350, 2, 1], First[#] + 2 == Last[#] &], PrimeQ] (* Robert G. Wilson v, Mar 12 2001 *)

Prog

(PARI) forprime(p=2, 14*10^3, my(q=p+2); if(isprime(q)&&isprime(p*q+2), print1(p*q+2", "))) \\ Bruce Nye, Apr 30 2026

Crossrefs

Cf. A048880, A000040, A001097, A001359, A005384, A006512, A037074, A054735.

Cf. A125251. - Zak Seidov, Dec 07 2008

Sequence in context: A153685 A208292 A121710 * A139579 A389110 A293206

Adjacent sequences: A051776 A051777 A051778 * A051780 A051781 A051782

Keyword

easy,nonn

Author

Joe DeMaio (jdemaio(AT)kennesaw.edu), Dec 09 1999

Extensions

Edited by R. J. Mathar, Aug 08 2008

Status

approved