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

Author

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

Keywords

• nonn
• easy

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

Examples

			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.
		

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

Crossrefs

Cf. A062326, A062718.

Programs

Magma

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

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 *)

Formula

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

Extensions

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