A230140 Number of ways to write n = x + y + z with 0 < x <= y <= z such that 6*x-1, 6*y-1, 6*z-1 are among those primes p (terms of A230138) with p + 2 and 2*p - 5 also prime.

0, 0, 1, 1, 2, 2, 3, 2, 3, 1, 2, 2, 2, 3, 3, 4, 2, 3, 2, 3, 3, 3, 4, 2, 5, 2, 6, 3, 6, 5, 4, 5, 3, 5, 5, 8, 7, 6, 5, 6, 5, 5, 7, 6, 8, 4, 6, 5, 6, 7, 9, 8, 8, 5, 7, 6, 8, 10, 6, 10, 4, 8, 6, 6, 10, 6, 9, 5, 6, 5, 7, 7, 9, 6, 7, 8, 5, 10, 6, 9, 6, 6, 7, 4, 7, 7, 9, 6, 5, 5, 4, 6, 5, 6, 5, 5, 6, 4, 6, 6

Offset: 1

Author

Zhi-Wei Sun, Oct 10 2013

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 2, i.e., 6*n-3 with n > 2 can be expressed as a sum of three terms from A230138. Moreover, for any integer n > 12, there are three distinct positive integers x, y, z with x + y + z = n such that 6*x-1, 6*y-1, 6*z-1 are primes in A230138.

(ii) For each integer n > 12, there are three distinct positive integers x, y, z with x + y + z = n such that 6*x-1, 6*y-1, 6*z-1 are among those primes p with p + 2 and 2*p + 9 also prime.

Note that part (i) of this conjecture implies that there are infinitely many primes in A230138.

Indices k such that a(m)>a(k) for all m>k, are (2, 10, 26, 334, 439, 544, 551, 684, ...). The only sequence which has the first 5 terms within the 3 lines of data is A212067. (Certainly a coincidence.) - M. F. Hasler, Oct 10 2013

Examples

			a(10) = 1 since 10 = 2 + 3 + 5, and the three numbers 6*2-1=11, 6*3-1=17 and 6*5-1=29 are terms of A230138.
		

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. A068307, A230138, A230141.

Programs

Mathematica

SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]&&PrimeQ[12n-7]
a[n_]:=Sum[If[SQ[i]&&SQ[j]&&SQ[n-i-j],1,0],{i,1,n/3},{j,i,(n-i)/2}]
Table[a[n],{n,1,100}]

PARI

ip(x)=isprime(6*x-1) && isprime(6*x+1) && isprime(12*x-7); a(n)=sum(x=1,n\3,sum(y=x,ip(x)*(n-x)\2,ip(y) && ip(n-x-y))) \\ - M. F. Hasler, Oct 10 2013