A124165 Primes p such that (2^p + 2^((p+1)/2) + 1)/5 is prime.

7, 17, 89, 1223, 5479, 11257, 11519, 12583, 23081, 36479, 52567, 52919, 125929, 365689, 1127239, 1148729, 4533073

Offset: 1

Author

Alexander Adamchuk, Dec 02 2006, Dec 04 2006

Keywords

• hard
• more
• nonn

Comments

PrimePi[ a(n) ] = {4, 7, 24, 200, 724, 1361, 1389, 1503, 2578, 3868, 5368, 5400, 11814, 31200, ...}.

3 terms found by David Broadhurst in Nov 2006: {36479, 52567, 52919}.

Only 2 terms found by Jean Penne in Nov 2006 belong to a(n): {125929, 365689}.

5 other numbers found by Jean Penne in Nov 2006 belong to related sequence of primes p such that (2^p - 2^((p+1)/2) + 1)/5 is prime: {221891, 235099, 305867, 311027, 333227}.

All terms belong to A124112 = {5, 7, 9, 11, 13, 17, 29, 43, 53, 89, 283, 557, 563, 613, 691, 1223, 2731, ...} (numbers k such that ((1+i)^k+1)/(2+i) is a Gaussian prime).

The terms 1127239 and 1148729 were found by Borys Jaworski in 2006-2007. - Alexander Adamchuk, Jun 20 2007

Links

• Henri Lifchitz and Renaud Lifchitz: PRP Records. Probable Primes Top 10000.

Crossrefs

Cf. A124112 (numbers k such that ((1+i)^k+1)/(2+i) is a Gaussian prime).

Programs

Mathematica

Do[p=Prime[n];f=(2^p+2^((p+1)/2)+1)/5;If[PrimeQ[f],Print[{n,p}]],{n,1,200}]

Extensions

a(17) from Serge Batalov, Mar 31 2014