

A147517


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


2



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

1,3


COMMENTS

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


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, pn1, if (isprime(2*pnp), 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



