A147517 Number of pairs of primes p < q such that (p+q)/2 = A002110(n), the n-th primorial.

0, 1, 6, 30, 190, 1564, 17075, 226758, 3792532, 82116003, 1975662890

Offset: 1

Author

Bill McEachen, Nov 05 2008

Keywords

• more
• nonn

Comments

The sequence is infinite and illustrates the number of primes expected to be centered around a given primorial.

Given ever-increasing primorial P, one can expect to find the highest symmetrical prime just below 2P.

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.

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

Examples

			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.
		

Links

• Bill McEachen, PARI script, Jun 03 2010

Crossrefs

Cf. A002110, A002375, A116979.

Programs

Mathematica

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 *)

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

Formula

a(n) = A002375(A002110(n)). - T. D. Noe, Nov 07 2008

Extensions

Better description by T. D. Noe, Nov 09 2008

Typo corrected typo by T. D. Noe, Nov 10 2008

Edited by Michel Marcus, Jul 09 2017

a(10)-a(11) from Bill McEachen, Jan 30 2018