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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 (list; graph; refs; listen; history; text; internal format)
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
CROSSREFS
Sequence in context: A216658 A214020 A029425 * A025902 A219923 A286950
KEYWORD
nonn,nice
AUTHOR
Zhi-Wei Sun, Nov 11 2012
STATUS
approved

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Last modified March 28 11:44 EDT 2024. Contains 371241 sequences. (Running on oeis4.)