

A091182


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


10



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
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OFFSET

1,5


COMMENTS

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 n1 = x+y1 <= 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 xy1 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 (m1)n is even, then we can write n as x+y, where x and y are positive integers with xym 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



EXAMPLE

a(8)=1 since 8=6+2 with 6*21 and 6*2+1 both prime.
a(11)=2 since 11=6+5=9+2 with 6*51, 6*5+1, 9*21, 9*2+1 all prime.


MAPLE

with(numtheory); a:=n>sum( (pi((i)*(ni)+1)  pi((i)*(ni)))*(pi((i)*(ni)1)  pi((i)*(ni)  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}] (* ZhiWei Sun, edited by T. D. Noe, Nov 29 2012 *)


CROSSREFS



KEYWORD

nonn,nice


AUTHOR



EXTENSIONS



STATUS

approved



