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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
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
Sequence in context: A259276 A109501 A239488 * A294221 A005922 A278008
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.)