OFFSET
1,6
COMMENTS
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.
(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.
(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.
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.
See also the comments of A229969 for other similar conjectures.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..4000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.
EXAMPLE
a(4) = 1 since 4 = 2+1+1 with 2*2-1 and 2*2+1 both prime.
a(5) = 1 since 5 = 3+1+1 with 2*3-1 and 2*3+1 both prime.
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.
MATHEMATICA
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}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 05 2013
STATUS
approved