cp's OEIS Frontend

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

%I A107317 #16 Sep 24 2018 16:53:14
%S A107317 6,14,26,62,86,146,314,422,482,614,842,926,1202,1514,2246,2966,3446,
%T A107317 5102,5942,6614,7082,7814,8846,9662,10226,11402,12014,12326,12962,
%U A107317 16022,16382,19802,20606,22262,24422,24866,27614,28562,34586,38366,40046
%N A107317 Semiprimes of the form 2*(m^2 + m + 1) (implying that m^2 + m + 1 is a prime).
%C A107317 Twice A002383.
%C A107317 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
%H A107317 Vincenzo Librandi, <a href="/A107317/b107317.txt">Table of n, a(n) for n = 1..300</a>
%F A107317 a(n) = 2*A002383(n).
%F A107317 a(n) = 2*(A002384(n)^2+A002384(n)+1).
%e A107317 a(1)=6 because 1^2 + 2^2 + 1 = 6 = 2*3;
%e A107317 a(2)=14 because 2^2 + 3^2 + 1 = 14 = 2*7;
%e A107317 a(3)=26 because 3^2 + 4^2 + 1 = 26 = 13*2.
%t A107317 2(#^2 + # + 1) & /@ Select[ Range[144], PrimeQ[ #^2 + # + 1] &] (* _Robert G. Wilson v_, May 28 2005 *)
%t A107317 fQ[n_] := Plus @@ Last /@ FactorInteger@n == 2 && IntegerQ@Sqrt[2n - 3]; Select[ Range@43513, fQ[ # ] &] (* _Robert G. Wilson v_ *)
%o A107317 (PARI) for(n=2,100000,if(bigomega(n)==2&&issquare(2*n-3),print1(n,","))) /* Lambert Herrgesell */
%Y A107317 Cf. A002383, A002384.
%K A107317 easy,nonn
%O A107317 1,1
%A A107317 _Giovanni Teofilatto_, May 21 2005
%E A107317 Edited by _Robert G. Wilson v_, May 28 2005
%E A107317 Re-edited by _N. J. A. Sloane_, Apr 18 2007