A091182 Number of ways to write n = x + y (x >= y > 0) with xy - 1 and xy + 1 both prime.

0, 0, 0, 1, 2, 0, 2, 1, 1, 0, 2, 0, 3, 0, 0, 1, 4, 1, 2, 0, 1, 1, 3, 1, 1, 0, 2, 2, 3, 0, 6, 3, 1, 1, 1, 1, 3, 4, 3, 0, 3, 1, 4, 2, 0, 1, 4, 1, 4, 1, 1, 3, 7, 0, 1, 1, 2, 1, 4, 0, 3, 2, 3, 4, 1, 0, 9, 2, 2, 1, 5, 1, 6, 1, 0, 3, 6, 1, 5, 0, 2, 3, 7, 1, 1, 2, 2

Offset: 1

Author

Ray Chandler, Dec 27 2003

Keywords

• nonn
• nice

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)

Sequence A090695 lists the 61 known values of n where a(n) = 0. - T. D. Noe, Nov 29 2012

Examples

			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.
		

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.

Crossrefs

Cf. A001359, A006512, A014574, A090695, A091183, A219157.

Programs

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 *)

Extensions

Edited by N. J. A. Sloane, Nov 29 2012