cp's OEIS Frontend

A229974 Number of ways to write n = x + y + z (x, y, z > 0) with the six numbers 2*x-1, 2*x+1, 2*x*y-1, 2*x*y+1, 2*x*y*z-1, 2*x*y*z+1 all prime.

%I A229974 #23 Oct 10 2013 03:00:32
%S A229974 0,0,0,1,1,4,2,1,2,4,5,3,3,8,1,9,4,6,3,8,16,8,4,8,7,3,10,7,3,14,4,6,8,
%T A229974 13,12,14,6,8,13,7,13,15,13,9,9,10,7,13,14,7,16,15,12,8,16,31,11,6,16,
%U A229974 13,16,15,26,8,10,17,10,12,11,17,9,9,13,18,17,23,14,10,7,13,29,13,18,14,9,19,21,14,19,14,25,11,14,18,13,21,15,26,14,8
%N A229974 Number of ways to write n = x + y + z (x, y, z > 0) with the six numbers 2*x-1, 2*x+1, 2*x*y-1, 2*x*y+1, 2*x*y*z-1, 2*x*y*z+1 all prime.
%C A229974 Conjecture: (i) a(n) > 0 for all n > 3. Moreover, any integer n > 3 can be written as x + y + z with x among 2, 3, 6 such that {2*x*y-1, 2*x*y+1} and {2*x*y*z-1, 2*x*y*z+1} are twin prime pairs.
%C A229974 (ii) Each integer n > 11 can be written as x + y + z (x, y, z > 0) with x-1, x+1, x*y-1, x*y+1, x*y*z-1, x*y*z+1 all prime, moreover we may require that x is among 4, 6, 12.
%C A229974 (iii) Any integer n > 3 not equal to 10 can be written as x + y + z (x, y, z > 0) such that the three numbers 2*x-1, 2*x*y-1 and 2*x*y*z-1 are Sophie Germain primes, moreover we may require that x is among 2, 3, 6.
%C A229974 Note that part (i) or (ii) of the above conjecture implies the twin prime conjecture, while part (iii) implies that there are infinitely many Sophie Germain primes.
%C A229974 See also the comments of A229969 for other similar conjectures.
%H A229974 Zhi-Wei Sun, <a href="/A229974/b229974.txt">Table of n, a(n)  for n = 1..4000</a>
%H A229974 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.
%e A229974 a(4) = 1 since 4 = 2+1+1 with 2*2-1 and 2*2+1 both prime.
%e A229974 a(5) = 1 since 5 = 3+1+1 with 2*3-1 and 2*3+1 both prime.
%e A229974 a(15) = 1 since 15 = 6+5+4 with 2*6-1, 2*6+1, 2*6*5-1, 2*6*5+1, 2*6*5*4-1, 2*6*5*4+1 all prime.
%t A229974 a[n_]:=Sum[If[PrimeQ[2i-1]&&PrimeQ[2i+1]&&PrimeQ[2*i*j-1]&&PrimeQ[2i*j+1]&&PrimeQ[2i*j*(n-i-j)-1]&&PrimeQ[2i*j*(n-i-j)+1],1,0],{i,1,n-2},{j,1,n-1-i}]
%t A229974 Table[a[n],{n,1,100}]
%Y A229974 Cf. A001359, A006512, A005384, A229969, A219842, A219864.
%K A229974 nonn
%O A229974 1,6
%A A229974 _Zhi-Wei Sun_, Oct 05 2013