This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A266948 #24 Nov 13 2019 20:55:52 %S A266948 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, %T A266948 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, %U A266948 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 %N A266948 Least prime p such that p-2 and 6n-p are also prime, or 0 if no such prime exists. %C A266948 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. %C A266948 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. %C A266948 This conjecture implies that a(n) > 0 for all n > 1. %C A266948 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], ... %C A266948 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. %H A266948 Harvey Dubner, <a href="/A007534/a007534.pdf">Twin Prime Conjectures</a>, Journal of Recreational Mathematics, Vol. 30 (3), 1999-2000. %H A266948 Bill Krys, <a href="https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/25782">Not much response but I still think this is outrageous result</a>, Yahoo! group primenumbers, Jan. 6, 2016 %H A266948 Bill Krys, <a href="/A266948/a266948.png">not much response, but i still think this is outrageous result</a>, message in primenumbers Yahoo group, Jan 6, 2016 [cached copy]. %H A266948 Dan Zwillinger, <a href="http://dx.doi.org/10.1090/S0025-5718-1979-0528060-5">A Goldbach Conjecture Using Twin Primes</a>, Math. Comp. 33, No.147 (1979), p.1071. %o A266948 (PARI) 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)) %K A266948 nonn %O A266948 0,3 %A A266948 _M. F. Hasler_, Jan 06 2016