A229969 - OEIS

%I #34 Oct 10 2013 03:50:42

%S 0,0,0,0,0,1,1,1,1,2,1,1,1,2,1,4,4,3,3,3,3,2,3,3,3,3,4,2,7,4,3,5,3,2,

%T 6,3,4,3,4,5,3,4,6,6,3,5,4,5,6,9,4,8,4,7,10,2,6,12,9,1,7,7,6,12,10,3,

%U 7,8,8,9,9,5,3,7,3,7,3,9,10,8,6,11,11,13,15,6,6,10,15,11,11,13,8,12,12,7,10,8,13,12

%N Number of ways to write n = x + y + z with 0 < x <= y <= z such that all the six numbers 2*x-1, 2*y-1, 2*z-1, 2*x*y-1, 2*x*z-1, 2*y*z-1 are prime.

%C Conjecture: a(n) > 0 for all n > 5. Moreover, any integer n > 6 can be written as x + y + z with x among 3, 4, 6, 10, 15 such that 2*y-1, 2*z-1, 2*x*y-1, 2*x*z-1, 2*y*z-1 are prime.

%C We have verified this conjecture for n up to 10^6. As (2*x-1)+(2*y-1)+(2*z-1) = 2*(x+y+z)-3, it implies Goldbach's weak conjecture which has been proved.

%C Zhi-Wei Sun also had some similar conjectures including the following (i)-(iii):

%C (i) Any integer n > 6 can be written as x + y + z (x, y, z > 0) with 2*x-1, 2*y-1, 2*z-1 and 2*x*y*z-1 all prime and x among 2, 3, 4. Also, each integer n > 2 can be written as x + y + z (x, y, z > 0) with 2*x+1, 2*y+1, 2*z+1 and 2*x*y*z+1 all prime and x among 1, 2, 3.

%C (ii) Each integer n > 4 can be written as x + y + z with x = 3 or 6 such that 2*y+1, 2*x*y*z-1 and 2*x*y*z+1 are prime.

%C (iii) Every integer n > 5 can be written as x + y + z (x, y, z > 0) with x*y-1, x*z-1, y*z-1 all prime and x among 2, 6, 10. Also, any integer n > 2 not equal to 16 can be written as x + y + z (x, y, z > 0) with x*y+1, x*z+1, y*z+1 all prime and x among 1, 2, 6.

%C See also A229974 for a similar conjecture involving three pairs of twin primes.

%H Zhi-Wei Sun, <a href="/A229969/b229969.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.

%e a(10) = 2 since 10 = 2+2+6 = 3+3+4 with 2*2-1, 2*6-1, 2*2*2-1, 2*2*6 -1, 2*3-1, 2*4-1, 2*3*3-1, 2*3*4-1 all prime.

%t a[n_]:=Sum[If[PrimeQ[2i-1]&&PrimeQ[2j-1]&&PrimeQ[2(n-i-j)-1]&&PrimeQ[2i*j-1]&&PrimeQ[2i(n-i-j)-1]&&PrimeQ[2j(n-i-j)-1],1,0],{i,1,n/3},{j,i,(n-i)/2}]

%t Table[a[n],{n,1,100}]

%Y Cf. A000040, A068307, A219842, A219864, A227923, A229974.

%K nonn

%O 1,10

%A _Zhi-Wei Sun_, Oct 04 2013