A137270 Primes p such that p^2 - 6 is also prime.

3, 5, 7, 13, 17, 23, 47, 53, 67, 73, 83, 97, 107, 113, 167, 193, 197, 263, 293, 317, 367, 373, 383, 457, 463, 467, 487, 503, 557, 593, 607, 643, 647, 673, 677, 683, 773, 787, 797, 823, 827, 857, 877, 887, 947, 1033, 1063, 1087, 1103, 1187, 1193, 1223, 1303

Offset

1,1

Comments

Each of the primes p = 2,3,5,7,13 has the property that the quadratic polynomial phi(x) = x^2 + x - p^2 takes on only prime values for x = 1,2,...,2p-2; each case giving exactly one repetition, in phi(p-1) = -p and phi(p) = p.

The only common term in A062718 and A137270 is 5. - Zak Seidov, Jun 16 2015

References

F. G. Frobenius, Uber quadratische Formen, die viele Primzahlen darstellen, Sitzungsber. d. Konigl. Acad. d. Wiss. zu Berlin, 1912, 966 - 980.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

A000040 INTERSECT A028879. - R. J. Mathar, Mar 16 2008

Example

The (2 x 7 - 2) -1 = 11 primes given by the polynomial x^2 + x - 7^2 for x = 1, 2, ..., 2 x 7 - 2 are -47, -43, -37, -29, -19, -7, 7, 23, 41, 61, 83, 107.

Maple

isA028879 := proc(n) isprime(n^2-6) ; end: isA137270 := proc(n) isprime(n) and isA028879(n) ; end: for i from 1 to 300 do if isA137270(ithprime(i)) then printf("%d, ", ithprime(i)) ; fi ; od: # R. J. Mathar, Mar 16 2008

Mathematica

Select[Prime[Range[2, 300]], PrimeQ[#^2-6]&] (* Harvey P. Dale, Jul 24 2012 *)

Prog

(Magma) [p: p in PrimesUpTo(1350) | IsPrime(p^2-6)]; // Vincenzo Librandi, Apr 14 2013

Crossrefs

Cf. A062326, A062718.

Sequence in context: A104294 A262100 A171566 * A071111 A177070 A247018

Adjacent sequences: A137267 A137268 A137269 * A137271 A137272 A137273

Keyword

nonn,easy

Author

Ben de la Rosa and Johan Meyer (meyerjh.sci(AT)ufa.ac.za), Mar 13 2008

Extensions

Corrected and extended by R. J. Mathar, Mar 16 2008

Status

approved