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A137270 Primes p such that p^2 - 6 is also prime. 4
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 (list; graph; refs; listen; history; text; internal format)
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.

Solutions of the equation n' + (n^2-6)' = 2, where n' is the arithmetic derivative of n. [Paolo P. Lava, Nov 09 2012]

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

f[n_]:=n^2-6; lst={}; Do[p=Prime[n]; If[PrimeQ[f[p]], AppendTo[lst, p]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 16 2009 *)

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

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Last modified June 2 17:02 EDT 2020. Contains 334787 sequences. (Running on oeis4.)