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%I #29 Jan 27 2021 10:39:00
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,1,1,1,1,3,1,1,2,2,1,3,1,1,2,2,
%T 1,3,1,0,2,2,1,2,2,1,2,1,1,2,1,2,2,2,2,3,1,1,3,2,1,4,1,0,3,3,1,3,1,1,
%U 3,3,1,2,2,2,2,2,2,3,1,3,3,1,2,6,1,2,2,1,3,5,0,1,4,2,1,4,0,1,4,3
%N Number of ways to write n = p+q(3-(-1)^n)/2 with p>q and p, q, p-6, q+6 all prime.
%C Conjecture: a(n) > 0 for all even n > 8012 and odd n > 15727.
%C This implies Goldbach's conjecture, Lemoine's conjecture and the conjecture that there are infinitely many primes p with p+6 also prime.
%C It has been verified for n up to 10^8.
%C Zhi-Wei Sun also made the following general conjecture: For any two multiples d_1 and d_2 of 6, all sufficiently large integers n can be written as p+q(3-(-1)^n)/2 with p>q and p, q, p-d_1, q+d_2 all prime. For example, for (d_1,d_2) = (-6,6),(-6,-6),(6,-6),(12,6),(-12,-6), it suffices to require that n is greater than 15721, 15733, 15739, 16349, 16349 respectively.
%H Zhi-Wei Sun, <a href="/A219055/b219055.txt">Table of n, a(n) for n = 1..100000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.
%e a(18) = 2 since 18 = 5+13 = 7+11 with 5+6, 13-6, 7+6, 11-6 all prime.
%t a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[n-(1+Mod[n,2])Prime[k]]==True&&PrimeQ[n-(1+Mod[n,2])Prime[k]-6]==True,1,0],{k,1,PrimePi[(n-1)/(2+Mod[n,2])]}]
%t Do[Print[n," ",a[n]],{n,1,100000}]
%o (PARI) A219055(n)={my(c=1+bittest(n, 0), s=0); forprime(q=1, (n-1)\(c+1), isprime(q+6) && isprime(n-c*q) && isprime(n-c*q-6) && s++); s} \\ _M. F. Hasler_, Nov 11 2012
%Y Cf. A023201, A002375, A046927, A218754, A218585, A218654, A218825, A219023, A219026, A219052.
%K nonn,nice
%O 1,18
%A _Zhi-Wei Sun_, Nov 11 2012
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