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%I #34 Feb 10 2014 02:06:24
%S 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,
%T 24,36,34,36,38,42,44,43,44,51,53,59,56,48,53,57,58,57,60,75,78,87,87,
%U 78,79,67,65
%N Number of ways of writing the n-th twin prime p as p = q + r + s, where q >= r >= s are twin primes.
%C The author conjectures that a(n) >= 1 for all n >= 4.
%C 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
%e 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.
%o (PARI) isA001097(n) = (isprime(n) & (isprime(n+2) | isprime(n-2)))
%o 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}
%o n=1; for( p=1, 700, if( isA001097(p), print(n, " ", A187754(p)); n=n+1)) /* _Michael B. Porter_, Jan 05 2013 */
%Y Cf. A001097.
%K nonn
%O 1,5
%A _Fabio Mercurio_, Jan 03 2013
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