A065017 Primes of the form p*q + p + q, where (p, q=p+2) are twin primes.

23, 47, 167, 359, 1847, 3719, 10607, 19319, 97967, 177239, 273527, 657719, 1042439, 1104599, 1329407, 1515359, 1745039, 2042039, 4464767, 5013119, 5148359, 9740639, 11095559, 11377127, 12538679, 16024007, 16410599, 16752647

Offset: 1

Author

Stephan Wagler (stephanwagler(AT)aol.com), Nov 01 2001

Keywords

• nonn

Comments

The resulting prime can never be a twin prime since the odd number preceding it is divisible by three and the following odd number is a perfect square.

Examples

			(3*5) + (3+5) = 23.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A049001, A049002.

Programs

Mathematica

NextPrim[n_] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; Return[k]]; k = 1; Do[k = NextPrim[k]; If[ PrimeQ[k + 2], p = k*(k + 2) + 2k + 2; If[ PrimeQ[p], Print[p]]], {n, 1, 700} ]
f[n_]:=Module[{x=Total[n]+Times@@n},If[PrimeQ[x],x,0]]; Select[f/@ (Select[Partition[Prime[Range[700]],2,1],Last[#]-First[#]==2&]), #!=0&] (* Harvey P. Dale, May 11 2011 *)

PARI

{ n=p=0; for (m=1, 10^9, p=nextprime(p + 1); if (isprime(q=p + 2) && isprime(a=p*q + p + q), write("b065017.txt", n++, " ", a); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 03 2009

Formula

p^2 + 4*p + 2.

Extensions

Offset changed from 0 to 1 by Harry J. Smith, Oct 03 2009