A266950 -id:A266950 - cp's OEIS frontend

A266948 Least prime p such that p-2 and 6n-p are also prime, or 0 if no such prime exists.

0, 0, 5, 5, 5, 7, 5, 5, 5, 7, 7, 5, 5, 5, 5, 7, 7, 5, 5, 5, 7, 13, 5, 7, 5, 13, 5, 5, 5, 7, 7, 5, 13, 5, 5, 13, 5, 31, 5, 5, 7, 5, 13, 7, 7, 7, 5, 5, 5, 13, 7, 13, 5, 5, 7, 13, 5, 5, 31, 5, 7, 7, 5, 5, 5, 7, 7, 5, 7, 5, 19, 5, 13, 5, 5, 7, 7, 5, 5, 7, 13, 7, 5, 7, 5, 7, 7, 13, 5, 13, 19, 5, 5, 109, 7, 7, 5, 5, 19, 7, 7, 5, 5, 5, 5, 13, 5, 43, 5, 7, 7, 5, 13, 5, 7, 7, 5, 19, 7, 5, 19

Offset: 0

M. F. Hasler, Jan 06 2016

nonn

Goldbach conjecture related: Group the consecutive even numbers in groups of three, (6n-2, 6n, 6n+2). The existence of a(n) corresponds to a Goldbach decomposition 6n = p + (6n-p) using the upper of a twin prime pair. Then 6n-2 = (p-2) + 6n-p is automatically a valid Goldbach decomposition of 6n-2, and 6n+2 = p + 6n+2-p is such a decomposition for 6n+2 if 6n+2-p (or 6n+4-p) is prime.
Zwillinger conjectured already in 1978 that for all n > 701 there is a p such that all these conditions are satisfied (not necessarily p = a(n)). See also A266952 - A266953.
This conjecture implies that a(n) > 0 for all n > 1.
See A266950 - A266951 for record values and indices. For easier reference we list some of these [n, a(n)] here: [21, 13]; [133, 139]; [1759, 241]; [10919, 643],  [112723, 1621]; [1072318, 2311], [1458993, 3001], [2617393, 3301], ...
Since a larger value of a(n) indicates that it was "difficult" to find a suitable twin prime p, this slow growth is a strong evidence that a(n) > 0 for all n > 1.

Harvey Dubner, Twin Prime Conjectures, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000.
Bill Krys, Not much response but I still think this is outrageous result, Yahoo! group primenumbers, Jan. 6, 2016
Bill Krys, not much response, but i still think this is outrageous result, message in primenumbers Yahoo group, Jan 6, 2016 [cached copy].
Dan Zwillinger, A Goldbach Conjecture Using Twin Primes, Math. Comp. 33, No.147 (1979), p.1071.

A266948(n)=my(GP(n,p=2)=forprime(p=p,n,isprime(n*2-p)&&return(p)));for(p=1,3*n,isprime(-2+p=GP(3*n,p))&&return(p))

A266951 Record values in A266948: least prime p such that p-2 and 6n-p are also prime.

Original entry on oeis.org

0, 5, 7, 13, 31, 109, 139, 199, 241, 283, 601, 619, 643, 823, 859, 883, 1021, 1051, 1093, 1291, 1621, 1873, 1879, 1951, 2311, 3001, 3301, 3361, 3529, 3583, 3919, 3931, 4339, 5419, 5479, 5851, 6361

Offset: 1

M. F. Hasler, Jan 06 2016

Goldbach conjecture related. The values of A266948(n) range theoretically between 2 and n. The smaller they are, the easier it was to find a suitable twin prime, and thus the more likely the conjecture "A266948(n) > 0 for all n > 1" holds. These record values vs. the respective indices A266950 yield an estimate for the growth of an upper bound.

Cf. A266948, A266950.

a(28)-a(37) from Pontus von Brömssen, Jan 06 2024

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A266948 Least prime p such that p-2 and 6n-p are also prime, or 0 if no such prime exists.

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A266951 Record values in A266948: least prime p such that p-2 and 6n-p are also prime.

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