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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.
13
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, 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. 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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 04 2012
STATUS
approved