OFFSET
1,5
COMMENTS
From Zhi-Wei Sun, Nov 27 2012: (Start)
Conjecture: a(n) > 0 for all n > 3120. This has been verified for n up to 5*10^7.
Note that if x >= y > 0 and x+y = n then n-1 = x+y-1 <= xy <= ((x+y)/2)^2 = n^2/4. So the conjecture implies that there are infinitely many twin primes.
For n=4,5,...,3120 we can write n = x+y (x >= y > 0) with xy-1 prime.
For each positive integer n <= 3120 different from 1,6,30,54, we can write n = x+y (x >= y > 0) with xy+1 prime.
More generally, we have the following conjecture: Let m be any positive integer. If n is sufficiently large and (m-1)n is even, then we can write n as x+y, where x and y are positive integers with xy-m and xy+m both prime. This general conjecture implies that for any positive even integer d there are infinitely many primes p and q with difference d. (End)
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..50000 (first 10000 terms from T. D. Noe)
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
EXAMPLE
a(8)=1 since 8=6+2 with 6*2-1 and 6*2+1 both prime.
a(11)=2 since 11=6+5=9+2 with 6*5-1, 6*5+1, 9*2-1, 9*2+1 all prime.
MAPLE
with(numtheory); a:=n->sum( (pi((i)*(n-i)+1) - pi((i)*(n-i)))*(pi((i)*(n-i)-1) - pi((i)*(n-i) - 2)) , i=1..floor(n/2) ); seq(a(k), k=1..100); # Wesley Ivan Hurt, Jan 21 2013
MATHEMATICA
Table[cnt = 0; Do[If[PrimeQ[k*(n - k) - 1] && PrimeQ[k*(n - k) + 1], cnt++], {k, n/2}]; cnt, {n, 100}] (* Zhi-Wei Sun, edited by T. D. Noe, Nov 29 2012 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Ray Chandler, Dec 27 2003
EXTENSIONS
Edited by N. J. A. Sloane, Nov 29 2012
STATUS
approved