A187754 Number of ways of writing the n-th twin prime p as p = q + r + s, where q >= r >= s are twin primes.

0, 0, 0, 1, 2, 3, 3, 6, 5, 8, 7, 7, 8, 8, 9, 10, 12, 14, 13, 15, 14, 21, 20, 20, 22, 22, 23, 23, 24, 36, 34, 36, 38, 42, 44, 43, 44, 51, 53, 59, 56, 48, 53, 57, 58, 57, 60, 75, 78, 87, 87, 78, 79, 67, 65

Offset: 1

Author

Fabio Mercurio, Jan 03 2013

Keywords

• nonn

Comments

The author conjectures that a(n) >= 1 for all n >= 4.

By Zhi-Wei Sun's conjecture related to A219157, for any positive integer n not among 1, 10, 430 we can write 6n-1 = p+2q = p+q+q with p,p-2,q,q+2 all prime, also for any integer n>702 we can write 6n+1 = 6(n-1)+7 = p+q+7 with p,p-2,q,q+2 all prime. Thus the author's conjecture is a consequence of Sun's conjecture. - Zhi-Wei Sun, Jan 06 2013

Examples

			a(9) = 5 because the ninth twin prime, A001097(9), is 31, and 31 can be written as a sum of three twin primes in 5 distinct ways: 3+11+17, 5+7+19, 5+13+13, 7+7+17, and 7+11+13.
		

Crossrefs

Cf. A001097.

Programs

PARI

isA001097(n) = (isprime(n) & (isprime(n+2) || isprime(n-2)))
A187754(n) = {local(q, r, s, a); a=0; for( q=1, n, if( isA001097(q), for( r=1, q, if( isA001097(r), for( s=1, r, if( isA001097(s) && (n==q+r+s), a=a+1)))))); a}
n=1; for( p=1, 700, if( isA001097(p), print(n, " ", A187754(p)); n=n+1)) /* Michael B. Porter, Jan 05 2013 */