lonely non-twin primes

[Neil Fernandez]

Hi,

I'd be very grateful to anyone with shedloads of available cycles who is
willing to help extend the sequence of n-tuply lonely non-twin primes. I
have tested up to p(10^4)=104729, and have found no higher terms than:

23, 1039

a(n), the first n-tuply lonely non-twin prime, is defined as the first
prime to be sandwiched between precisely n pairs of twin primes on each
side. The twin pairs do not have to belong to a quadruplet
(6q-1, 6q+1, 6q+5, 6q+7,...) (see A007530) or greater k-tuplet. All that
is necessary is that between the pairs there are no other primes. E.g.
a(2) = 1039 because the prime sequence runs
1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,
i.e. ...,non-twin, two twin pairs, 1039, two twin pairs, non-twin,...

A068016 is the sequence of lonely non-twin primes, i.e. sandwiched
between at least one twin pair on each side:
23, 37, 67, 233, 277, 631, 1039, 1283, 1297, 1307, 1613, 1693, 1709,
2099, 2137, 2333,2719, 2797, 3271, 3533, 3547, 3571, 3923, 4027, 4253,
4523, 4643, 4793, 5483, 5507, 5647, 6563, 7321,...

A069456 is the sequence of non-twin primes that are at least doubly
lonely, i.e. sandwiched between at least two twin pairs on each side:
1039,2099,4253,91121

There are no non-twins < 10^5 which are triply or higher-tuply lonely.

Many thanks :-)

Best regards,

Neil
-- 
Neil Fernandez