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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)



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.

Thus the number of pairs ~ exp(Y). For example, the fit yields exp(7.370)=1587 prime pairs for pf=6 (30030) while actual=1564. Gnumeric spreadsheet was used for data analysis.


Table of n, a(n) for n=1..11.

Bill McEachen, PARI script, Jun 03 2010


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


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.


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


Cf. A002110, A002375, A116979.

Sequence in context: A259276 A109501 A239488 * A294221 A005922 A278008

Adjacent sequences:  A147514 A147515 A147516 * A147518 A147519 A147520




Bill McEachen, Nov 05 2008


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



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Last modified December 5 09:03 EST 2020. Contains 338944 sequences. (Running on oeis4.)