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Number of ways to write n=p+q(3+(-1)^n)/2 with q<=n/2 and p, q, p^2+3pq+q^2 all prime.
13

%I #25 Sep 25 2023 12:47:18

%S 0,0,0,0,0,0,1,1,1,0,1,1,1,1,1,1,3,2,1,1,1,2,2,1,3,1,3,1,1,2,1,0,3,3,

%T 2,3,3,0,3,0,3,2,1,1,4,1,2,2,1,2,0,2,2,2,3,0,4,1,1,2,0,1,2,3,5,0,2,1,

%U 3,4,1,1,2,2,6,2,2,4,1,2,3,2,3,3,3,2,4,1,2,5,0,3,4,2,3,4,3,1,4,3

%N Number of ways to write n=p+q(3+(-1)^n)/2 with q<=n/2 and p, q, p^2+3pq+q^2 all prime.

%C Conjecture: a(n)>0 for all n>=1188.

%C Conjecture verified for n up to 10^9. - _Mauro Fiorentini_, Sep 23 2023

%C This conjecture is stronger than both Goldbach's conjecture and Lemoine's conjecture.

%C _Zhi-Wei Sun_ also made the following conjecture: Given any positive odd integer d, there is a prime p(d) such that for any prime p>p(d) there is a prime q<p such that p^2+dpq+q^2 is prime. For example, we may take p(1)=5, p(3)=2, p(5)=61, p(7)=3, p(9)=13, p(11)=7, p(13)=3, p(15)=163, p(17)=13, p(19)=5, p(21)=p(23)=2, p(25)=89, p(27)=3, p(29)=53.

%C Conjecture verified for d up to 100 and p up to 10^7. - _Mauro Fiorentini_, Sep 23 2023

%H Zhi-Wei Sun, <a href="/A218754/b218754.txt">Table of n, a(n) for n = 1..20000</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 For n=72 we have a(72)=1 since the only primes p and q with p+q=72, q<=36 and p^2+3pq+q^2 prime are p=67 and q=5.

%t a[n_]:=a[n]=Sum[If[PrimeQ[q]==True&&PrimeQ[n-q(3-(-1)^n)/2]&&PrimeQ[q^2+3q(n-q(3-(-1)^n)/2)+(n-q(3-(-1)^n)/2)^2]==True,1,0],{q,1,n/2}]

%t Do[Print[n," ",a[n]],{n,1,20000}]

%Y Cf. A002372, A046927, A218585, A218654, A218656.

%Y Cf. A000034 = 1,2,1,2,... = (3-(-1)^n)/2. (Note: Offset shifted w.r.t. use in the definition of this sequence.) - _M. F. Hasler_, Nov 05 2012

%K nonn

%O 1,17

%A _Zhi-Wei Sun_, Nov 04 2012