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 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. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). LINKS Table of n, a(n) for n=0..94. 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. PROG (PARI) 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)) CROSSREFS Cf. A007534, A266948, A266953. Sequence in context: A011472 A246506 A001733 * A245423 A242889 A092076 Adjacent sequences: A266949 A266950 A266951 * A266953 A266954 A266955 KEYWORD nonn AUTHOR M. F. Hasler, Jan 06 2016 STATUS approved

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Last modified September 11 02:28 EDT 2024. Contains 375813 sequences. (Running on oeis4.)