A107317 Semiprimes of the form 2*(m^2 + m + 1) (implying that m^2 + m + 1 is a prime).

6, 14, 26, 62, 86, 146, 314, 422, 482, 614, 842, 926, 1202, 1514, 2246, 2966, 3446, 5102, 5942, 6614, 7082, 7814, 8846, 9662, 10226, 11402, 12014, 12326, 12962, 16022, 16382, 19802, 20606, 22262, 24422, 24866, 27614, 28562, 34586, 38366, 40046

Offset: 1

Author

Giovanni Teofilatto, May 21 2005

Keywords

• easy
• nonn

Comments

Twice A002383.

Also semiprimes n such that 2*n - 3 is a square. - Giovanni Teofilatto, Dec 29 2005. This coincidence was noticed by Andrew S. Plewe. Proof that this is the same sequence: If X is n^2+(n+1)^2+1, then 2X-3 is 4n^2+4n+1 = (2n+1)^2. And if 2X-3 is a square, then since it's odd, 2X-3 = (2n+1)^2 and X = n^2+(n+1)^2+1. - Don Reble, Apr 18 2007

Examples

			a(1)=6 because 1^2 + 2^2 + 1 = 6 = 2*3;
a(2)=14 because 2^2 + 3^2 + 1 = 14 = 2*7;
a(3)=26 because 3^2 + 4^2 + 1 = 26 = 13*2.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A002383, A002384.

Programs

Mathematica

2(#^2 + # + 1) & /@ Select[ Range[144], PrimeQ[ #^2 + # + 1] &] (* Robert G. Wilson v, May 28 2005 *)
fQ[n_] := Plus @@ Last /@ FactorInteger@n == 2 && IntegerQ@Sqrt[2n - 3]; Select[ Range@43513, fQ[ # ] &] (* Robert G. Wilson v *)

PARI

for(n=2,100000,if(bigomega(n)==2&&issquare(2*n-3),print1(n,","))) /* Lambert Herrgesell */

Formula

a(n) = 2*A002383(n).

a(n) = 2*(A002384(n)^2+A002384(n)+1).

Extensions

Edited by Robert G. Wilson v, May 28 2005

Re-edited by N. J. A. Sloane, Apr 18 2007