

A107317


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


2



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
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OFFSET

1,1


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 2X3 is 4n^2+4n+1 = (2n+1)^2. And if 2X3 is a square, then since it's odd, 2X3 = (2n+1)^2 and X = n^2+(n+1)^2+1.  Don Reble, Apr 18 2007


LINKS

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


FORMULA

a(n) = 2*A002383(n).
a(n) = 2*(A002384(n)^2+A002384(n)+1).


EXAMPLE

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.


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 *)


PROG

(PARI) for(n=2, 100000, if(bigomega(n)==2&&issquare(2*n3), print1(n, ", "))) /* Lambert Herrgesell */


CROSSREFS

Cf. A002383, A002384.
Sequence in context: A131951 A168648 A093776 * A071776 A063590 A128806
Adjacent sequences: A107314 A107315 A107316 * A107318 A107319 A107320


KEYWORD

easy,nonn


AUTHOR

Giovanni Teofilatto, May 21 2005


EXTENSIONS

Edited by Robert G. Wilson v, May 28 2005
Reedited by N. J. A. Sloane, Apr 18 2007


STATUS

approved



