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%I #29 Jul 13 2022 15:54:01
%S 0,1,6,30,190,1564,17075,226758,3792532,82116003,1975662890
%N Number of pairs of primes p < q such that (p+q)/2 = A002110(n), the n-th primorial.
%C The sequence is infinite and illustrates the number of primes expected to be centered around a given primorial.
%C Given ever-increasing primorial P, one can expect to find the highest symmetrical prime just below 2P.
%C Using a limited dataset, the approximate relation is the quadratic Y=Ax^2+Bx+C (A,B,C)=(0.12267, 0.75758, -1.592) where Y = log(number of prime pairs) (each > the prime factors) and x is number of prime factors of the seed primorial.
%C Standard heuristics give a(n) ~ exp(gamma)*log(p)*p#/p^2 where p is the n-th prime and gamma is A001620. - _Charles R Greathouse IV_, Jul 13 2022
%H Bill McEachen, <a href="http://billymac00.pbworks.com/f/mirrors.gp">PARI script</a>, Jun 03 2010
%F a(n) = A002375(A002110(n)). - _T. D. Noe_, Nov 07 2008
%e There are 6 pairs centered at primorial=30: (29,31),(23,37),(19,41),(17,43),(13,47),(7,53). As they are symmetrical, each prime pair sums to twice the primorial center.
%t f = Compile[{{n, _Integer}}, Block[{p = 2, c = 0, pn = Times @@ Prime@ Range@ n}, While[p < pn, If[PrimeQ[ 2pn -p], c++]; p = NextPrime@ p]; c]]; Array[f, 10] (* _Robert G. Wilson v_, Feb 08 2018 *)
%o (PARI) a(n) = pn = prod(k=1, n, prime(k)); nb = 0; forprime(p=2, pn-1, if (isprime(2*pn-p), nb++)); nb; \\ _Michel Marcus_, Jul 09 2017
%Y Cf. A002110, A002375, A116979.
%K more,nonn
%O 1,3
%A _Bill McEachen_, Nov 05 2008
%E Better description by _T. D. Noe_, Nov 09 2008
%E Typo corrected typo by _T. D. Noe_, Nov 10 2008
%E Edited by _Michel Marcus_, Jul 09 2017
%E a(10)-a(11) from _Bill McEachen_, Jan 30 2018
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