A144094 -id:A144094 - cp's OEIS frontend

A266952 Least prime p such that p-2 and 6n-p and 6n+2-p are also prime, or 0 if no such prime exists.

0, 0, 7, 7, 7, 13, 7, 13, 7, 13, 19, 7, 13, 7, 13, 19, 0, 31, 7, 7, 13, 19, 31, 31, 7, 13, 7, 13, 19, 73, 31, 7, 13, 7, 7, 13, 19, 31, 31, 7, 13, 7, 13, 19, 73, 31, 7, 13, 7, 13, 19, 109, 31, 7, 13, 19, 109, 31, 109, 7, 13, 19, 61, 31, 73, 43, 199, 0, 61, 103, 73, 7, 13, 7, 13, 19, 109, 31, 7, 13, 19, 139, 31, 151, 43, 199, 0, 61, 7, 13, 19, 199, 31, 139, 43

Offset: 0

M. F. Hasler, Jan 06 2016

nonn

If a(n) > 0, then the triple {6n-2, 6n, 6n+2} of consecutive even numbers allows a "simultaneous Goldbach decomposition" using two pairs of twin primes, 6n-2 = p-2 + 6n-p ; 6n = p + 6n-p ; 6n+2 = p + 6n+2-p.
Up to 10^5, the only indices for which a(n)=0 are {0, 1, 16, 67, 86, 131, 151, 186, 191, 211, 226, 541, 701}. I conjecture that this list is finite, and probably complete. Is it a coincidence that all odd numbers > 1 in this list are primes? (See also A144094.)
This seems equivalent to a conjecture Zwillinger made in 1978, see reference in LINKS.
See A266953 for another variant with a slightly relaxed condition (instead of 6n+2-p one can also have 6n+4-p prime, but this affects only n=2 and n=67), and A266948 for another variant with less restrictive conditions (only p-2 and 6n-p have to be prime).

Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
Dan Zwillinger, A Goldbach Conjecture Using Twin Primes, Math. Comp. 33, No.147 (1979), p.1071.

Cf. A007534, A266948, A266953.

A266952(n)=my(GP(n, p=2)=forprime(p=p, n+1, isprime(n*2-p)&&return(p))); for(p=1, 3*n, isprime(-2+p=GP(3*n, p))+!p&&(!p||isprime(6*n+2-p))&&return(p))

cp's OEIS Frontend

A266952 Least prime p such that p-2 and 6n-p and 6n+2-p are also prime, or 0 if no such prime exists.

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