A266952 -id:A266952 - 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))

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.

Original entry on oeis.org

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

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 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?

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))

A144094 Prime numbers r such that there is no pair of twin primes p,p+2 and q,q+2 such that p+q+2 = 6r.

Original entry on oeis.org

67, 131, 151, 191, 211, 541, 701

Offset: 1

Pierre CAMI, Sep 10 2008

nonn

Probably the sequence is complete.

More generally, {1, 2, 16, 66, 67, 86, 116, 131, 151, 186, 191, 211, 226, 541, 701} seem to be the only numbers such that {6n-2, 6n, 6n+2} do not have a Goldbach type of decomposition (sum of two primes) using only two pairs of twin primes. See also A266952, A266953, and A266948 and A007534. - M. F. Hasler, Jan 07 2016

			6*3=18, 5,7 twin primes as 11,13, 5+13=7+11=18 so 3 not in the sequence 6*5=30, 11,13 twin primes as 17,19, 13+19=11+17=30 so 5 not in the sequence

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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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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.

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A144094 Prime numbers r such that there is no pair of twin primes p,p+2 and q,q+2 such that p+q+2 = 6r.

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