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.

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, Table of n, a(n) for n = 1..100000

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

Crossrefs

Cf. A023201, A002375, A046927, A218754, A218585, A218654, A218825, A219023, A219026, A219052.

Sequence in context: A216658 A214020 A029425 * A025902 A219923 A286950

Adjacent sequences: A219052 A219053 A219054 * A219056 A219057 A219058

Keyword

nonn,nice

Author

Zhi-Wei Sun, Nov 11 2012

Status

approved