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

1,5

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)

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

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

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

Sequence in context: A230417 A287263 A102442 * A001822 A351380 A346730

Adjacent sequences: A091179 A091180 A091181 * A091183 A091184 A091185

Keyword

nonn,nice

Author

Ray Chandler, Dec 27 2003

Extensions

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

Status

approved