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 A147517 Number of pairs of primes p < q such that (p+q)/2 = A002110(n), the n-th primorial. 2
 0, 1, 6, 30, 190, 1564, 17075, 226758, 3792532, 82116003, 1975662890 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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 LINKS Table of n, a(n) for n=1..11. Bill McEachen, PARI script, Jun 03 2010 FORMULA a(n) = A002375(A002110(n)). - T. D. Noe, Nov 07 2008 EXAMPLE 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. 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 *) PROG (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 CROSSREFS Cf. A002110, A002375, A116979. Sequence in context: A259276 A109501 A239488 * A294221 A005922 A278008 Adjacent sequences: A147514 A147515 A147516 * A147518 A147519 A147520 KEYWORD more,nonn AUTHOR Bill McEachen, Nov 05 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 STATUS approved

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Last modified May 20 10:16 EDT 2024. Contains 372710 sequences. (Running on oeis4.)