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%I #44 Oct 30 2023 07:44:33
%S 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,
%T 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,
%U 2,1,5,1,6,1,0,3,6,1,5,0,2,3,7,1,1,2,2
%N Number of ways to write n = x + y (x >= y > 0) with xy - 1 and xy + 1 both prime.
%C From _Zhi-Wei Sun_, Nov 27 2012: (Start)
%C Conjecture: a(n) > 0 for all n > 3120. This has been verified for n up to 5*10^7.
%C 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.
%C For n=4,5,...,3120 we can write n = x+y (x >= y > 0) with xy-1 prime.
%C 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.
%C 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)
%C Sequence A090695 lists the 61 known values of n where a(n) = 0. - _T. D. Noe_, Nov 29 2012
%H Zhi-Wei Sun, <a href="/A091182/b091182.txt">Table of n, a(n) for n = 1..50000</a> (first 10000 terms from T. D. Noe)
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e a(8)=1 since 8=6+2 with 6*2-1 and 6*2+1 both prime.
%e a(11)=2 since 11=6+5=9+2 with 6*5-1, 6*5+1, 9*2-1, 9*2+1 all prime.
%p 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
%t 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 *)
%Y Cf. A001359, A006512, A014574, A090695, A091183, A219157.
%K nonn,nice
%O 1,5
%A _Ray Chandler_, Dec 27 2003
%E Edited by _N. J. A. Sloane_, Nov 29 2012
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