A229969 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.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Oct 04 2013

Keywords

• nonn

Comments

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.

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.

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

(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.

(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.

(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.

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

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Crossrefs

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

Programs

Mathematica

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