A218825 - OEIS

%I #25 Jul 21 2023 08:41:27

%S 0,0,0,0,0,0,0,0,3,1,1,3,1,3,3,1,2,2,1,2,3,1,2,3,1,2,2,1,3,1,1,3,3,4,

%T 3,1,2,5,3,1,3,2,4,3,3,1,7,4,1,5,3,5,8,4,3,4,3,3,5,4,4,3,2,3,5,3,5,7,

%U 3,2,9,4,4,6,3,3,8,6,1,4,5,2,7,1,4,2,4,5,5,2,4,4,3,2,5,4,5,6,4,1

%N Number of ways to write 2n-1 as p+2q with p, q and p^2+60q^2 all prime.

%C Conjecture: a(n)>0 for all n>8.

%C This conjecture is stronger than Lemoine's conjecture. It has been verified for n up to 10^8.

%C Conjecture verified for 2n-1 up to 10^9. - _Mauro Fiorentini_, Jul 20 2023

%C Zhi-Wei Sun also made the following general conjecture: For any positive integer n, the set E(n) of positive odd integers not of the form p+2q with p, q, p^2+4(2^n-1)q^2 all prime, is finite. In particular, if we let M(n) denote the maximal element of E(n), then M(1)=3449, M(2)=1711, E(3)={1,3,5,7,31,73}, E(4)={1,3,5,7,9,11,13,15},

%C   M(5)=6227, M(6)=1051, M(7)=2239, M(8)=2599, M(9)=7723,

%C   M(10)=781, M(11)=1163, M(12)=587, M(13)=11443,

%C   M(14)=2279, M(15)=157, M(16)=587, M(17)=32041,

%C   M(18)=1051, M(19)=2083, M(20)=4681.

%C Conjecture verified for 2n-1 up to 10^9 for n <= 4 and up to 10^6 for n <= 20. - _Mauro Fiorentini_, Jul 20 2023

%C Zhi-Wei Sun also guessed that for any positive even integer d not congruent to 2 modulo 6 there exists a prime p(d) such that for any prime p>p(d) there is a prime q<p with p^2+dq^2 prime. In particular, we may take

%C   p(4)=p(6)=3, p(10)=5, p(12)=3, p(16)=2, p(18)=3,

%C   p(22)=11, p(24)=17, and p(28)=p(30)=7.

%H Zhi-Wei Sun, <a href="/A218825/b218825.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.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lemoine%27s_conjecture">Lemoine's conjecture</a>

%e a(10)=1 since the only primes p and q with p^2+60q^2 prime and p+2q=19 are p=13 and q=3.

%t a[n_]:=a[n]=Sum[If[PrimeQ[q]==True&&PrimeQ[2n-1-2q]==True&&PrimeQ[(2n-1-2q)^2+60q^2]==True,1,0],{q,1,n-1}]

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

%o (PARI) A218825(n)={my(c=0,n21=n*2-1);forprime(q=2,n-1,isprime(n21-2*q) || next; isprime(q^2*60+(n21-2*q)^2) && c++); c}  \\ _M. F. Hasler_, Nov 07 2012

%Y Cf. A000040, A046927, A218754, A218585, A218654, A218656, A218797.

%K nonn

%O 1,9

%A _Zhi-Wei Sun_, Nov 07 2012