A266953 - OEIS

A266953

Least prime p such that p-2 and 6n-p and either 6n+2-p or 6n+4-p is also prime, or 0 if no such prime exists.

0, 0, 5, 5, 5, 13, 7, 5, 5, 13, 19, 7, 5, 7, 5, 19, 0, 5, 5, 5, 13, 19, 5, 31, 7, 13, 7, 13, 5, 73, 31, 7, 13, 5, 7, 13, 19, 31, 5, 5, 13, 7, 13, 19, 73, 31, 7, 5, 7, 13, 19, 109, 5, 5, 13, 19, 109, 31, 109, 5, 13, 19, 61, 31, 5, 43, 199, 5, 61, 103, 73, 7, 13, 7, 5, 19, 109, 5, 5, 13, 19, 139, 5, 151, 5, 199, 0, 61, 7, 13, 19, 199, 31, 139, 43, 109, 7, 13, 19

(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 only 4 different primes, 6n-2 = p-2 + 6n-p ; 6n = p + 6n-p ; 6n+2 = p + 6n+2-p = p-2 + 6n+4-p.

See A266952 for the version which does not allow the second decomposition of the last member. See A266948 for a variant which does not require 6n+2-p to be prime.

Up to 10^5, the only indices for which a(n)=0 are {0, 1, 16, 86, 131, 151, 186, 191, 211, 226, 541, 701}. (Only 2 and 67 require the alternative primality of 6n+4-p and have thus A266952(n)=0.) I conjecture that this list is finite, and probably complete. Is it a coincidence that all odd numbers in this list are primes?

LINKS

Table of n, a(n) for n=0..98.

PROG

(PARI) A266953(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)||isprime(6*n+4-p))&&return(p))

CROSSREFS

Sequence in context: A119991 A262947 A265821 * A224165 A330318 A334934

Adjacent sequences: A266950 A266951 A266952 * A266954 A266955 A266956

KEYWORD

nonn

AUTHOR

M. F. Hasler, Jan 06 2016

STATUS

approved