|
%I #20 Oct 14 2013 08:18:11
%S 0,1,2,3,2,3,4,4,4,3,4,1,3,3,3,5,5,4,3,6,4,7,7,2,4,6,4,4,6,3,1,4,2,4,
%T 7,4,1,4,4,2,6,4,3,4,2,3,5,3,2,1,2,3,6,2,6,6,3,5,4,5,3,7,2,4,6,2,4,5,
%U 3,5,8,5,2,10,4,4,8,5,6,7,8,4,11,4,3,6,4,2,4,8,8,11,5,3,11,5,3,6,4,5
%N Number of ways to write n = x + y (x, y > 0) with 2*x + 1, x^2 + x + 1 and y^2 + y + 1 all prime.
%C Conjecture: (i) a(n) > 0 for all n > 1.
%C (ii) Any integer n > 3 can be written as p + q with p, 2*p - 3 and q^2 + q + 1 all prime. Also, each integer n > 3 not equal to 30 can be expressed as p + q with p, q^2 + q - 1 and q^2 + q + 1 all prime.
%C (iii) Any integer n > 1 can be written as x + y (x, y > 0) with x^2 + 1 (or 4*x^2+1) and y^2 + y + 1 (or 4*y^2 + 1) both prime.
%C (iv) Each integer n > 3 can be expressed as p + q (q > 0) with p, 2*p - 3 and 4*q^2 + 1 all prime.
%C (v) Any even number greater than 4 can be written as p + q with p, q and p^2 + 4 (or p^2 - 2) all prime. Also, each even number greater than 2 and not equal to 122 can be expressed as p + q with p, q and (p-1)^2 + 1 all prime.
%C We have verified the first part for n up to 10^8.
%H Zhi-Wei Sun, <a href="/A230252/b230252.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.
%e a(5) = 2 since 5 = 2 + 3 = 3 + 2, and 2*2+1 = 5, 2*3+1 = 7, 2^2+2+1 = 7, 3^2+3+1 = 13 are all prime.
%e a(31) = 1 since 31 = 14 + 17, and 2*14+1 = 29, 14^2+14+1 = 211 and 17^2+17+1 = 307 are all prime.
%t a[n_]:=Sum[If[PrimeQ[2i+1]&&PrimeQ[i^2+i+1]&&PrimeQ[(n-i)^2+n-i+1],1,0],{i,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A002383, A002384, A002375, A219864, A227908, A227909, A230230, A230241, A230254.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Oct 13 2013
|
