|
|
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.
|
|
LINKS
|
|
|
FORMULA
|
|
|
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
|
|
|
KEYWORD
|
more,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Typo corrected typo by T. D. Noe, Nov 10 2008
|
|
STATUS
|
approved
|
|
|
|