login
A219055
Number of ways to write n = p+q(3-(-1)^n)/2 with p>q and p, q, p-6, q+6 all prime.
15
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 2, 2, 1, 3, 1, 0, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 1, 3, 2, 1, 4, 1, 0, 3, 3, 1, 3, 1, 1, 3, 3, 1, 2, 2, 2, 2, 2, 2, 3, 1, 3, 3, 1, 2, 6, 1, 2, 2, 1, 3, 5, 0, 1, 4, 2, 1, 4, 0, 1, 4, 3
OFFSET
1,18
COMMENTS
Conjecture: a(n) > 0 for all even n > 8012 and odd n > 15727.
This implies Goldbach's conjecture, Lemoine's conjecture and the conjecture that there are infinitely many primes p with p+6 also prime.
It has been verified for n up to 10^8.
Zhi-Wei Sun also made the following general conjecture: For any two multiples d_1 and d_2 of 6, all sufficiently large integers n can be written as p+q(3-(-1)^n)/2 with p>q and p, q, p-d_1, q+d_2 all prime. For example, for (d_1,d_2) = (-6,6),(-6,-6),(6,-6),(12,6),(-12,-6), it suffices to require that n is greater than 15721, 15733, 15739, 16349, 16349 respectively.
LINKS
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.
EXAMPLE
a(18) = 2 since 18 = 5+13 = 7+11 with 5+6, 13-6, 7+6, 11-6 all prime.
MATHEMATICA
a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+6]==True&&PrimeQ[n-(1+Mod[n, 2])Prime[k]]==True&&PrimeQ[n-(1+Mod[n, 2])Prime[k]-6]==True, 1, 0], {k, 1, PrimePi[(n-1)/(2+Mod[n, 2])]}]
Do[Print[n, " ", a[n]], {n, 1, 100000}]
PROG
(PARI) A219055(n)={my(c=1+bittest(n, 0), s=0); forprime(q=1, (n-1)\(c+1), isprime(q+6) && isprime(n-c*q) && isprime(n-c*q-6) && s++); s} \\ M. F. Hasler, Nov 11 2012
KEYWORD
nonn,nice
AUTHOR
Zhi-Wei Sun, Nov 11 2012
STATUS
approved