A218825 Number of ways to write 2n-1 as p+2q with p, q and p^2+60q^2 all prime.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Nov 07 2012

Keywords

• nonn

Comments

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

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

Conjecture verified for 2n-1 up to 10^9. - Mauro Fiorentini, Jul 20 2023

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},

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

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

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

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

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

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

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

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.
• Wikipedia, Lemoine's conjecture

Crossrefs

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

Programs

Mathematica

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

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

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