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%I #23 Feb 03 2024 00:51:08
%S 3,5,7,13,17,23,47,53,67,73,83,97,107,113,167,193,197,263,293,317,367,
%T 373,383,457,463,467,487,503,557,593,607,643,647,673,677,683,773,787,
%U 797,823,827,857,877,887,947,1033,1063,1087,1103,1187,1193,1223,1303
%N Primes p such that p^2 - 6 is also prime.
%C 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.
%C The only common term in A062718 and A137270 is 5. - _Zak Seidov_, Jun 16 2015
%D F. G. Frobenius, Uber quadratische Formen, die viele Primzahlen darstellen, Sitzungsber. d. Konigl. Acad. d. Wiss. zu Berlin, 1912, 966 - 980.
%H Vincenzo Librandi, <a href="/A137270/b137270.txt">Table of n, a(n) for n = 1..1000</a>
%F A000040 INTERSECT A028879. - _R. J. Mathar_, Mar 16 2008
%e 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.
%p 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
%t Select[Prime[Range[2,300]],PrimeQ[#^2-6]&] (* _Harvey P. Dale_, Jul 24 2012 *)
%o (Magma) [p: p in PrimesUpTo(1350) | IsPrime(p^2-6)]; // _Vincenzo Librandi_, Apr 14 2013
%Y Cf. A062326, A062718.
%K nonn,easy
%O 1,1
%A Ben de la Rosa and Johan Meyer (meyerjh.sci(AT)ufa.ac.za), Mar 13 2008
%E Corrected and extended by _R. J. Mathar_, Mar 16 2008
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