A218754 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.

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, 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, 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

Offset

1,17

Comments

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

Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 23 2023

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

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.

Conjecture verified for d up to 100 and p up to 10^7. - Mauro Fiorentini, Sep 23 2023

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.

Example

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.

Mathematica

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

Crossrefs

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

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

Sequence in context: A093032 A072115 A210650 * A079948 A257657 A339584

Adjacent sequences: A218751 A218752 A218753 * A218755 A218756 A218757

Keyword

nonn

Author

Zhi-Wei Sun, Nov 04 2012

Status

approved