A220091 Number of ways to write n=p+q+(n mod 2)q with p>q and p, q, 6q-1, 6q+1 all prime

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 3, 1, 2, 1, 1, 1, 3, 2, 2, 3, 2, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 3, 1, 2, 3, 2, 3, 3, 1, 1, 4, 2, 1, 3, 1, 1, 3, 4, 3, 3, 2, 1, 1, 3, 3, 1, 2, 2, 4, 4, 5, 3, 1, 1, 3, 2, 3, 3, 2, 2, 4, 2, 3, 3, 0, 1, 5, 2, 2, 3, 1, 0, 2, 3

Offset

1,11

Comments

Conjecture: a(n)>0 for all even n>=8070 and odd n>=18680.

This conjecture unifies the twin prime conjecture, Goldbach's conjecture and Lemoine's conjecture. It has been verified for n up to 10^7.

Zhi-Wei Sun also made the following conjecture: Any integer n>=6782 can be written as p+q+(n mod 2)q with p>q and p, q, q-6, q+6 all prime, and any integer n>=4410 can be written as p+q+(n mod 2)q with p>q and p, q, 2q-3, 2q+3 all prime, and any integer n>=16140 can be written as p+q+(n mod 2)q with p>q and p, q, 3q-2, 3q+2 all prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Example

a(31)=1 since 31=17+2*7 with 6*7-1 and 6*7+1 twin primes.
a(32)=1 since 32=29+3 with 6*3-1 and 6*3+1 twin primes.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[6Prime[k]-1]==True&&PrimeQ[6Prime[k]+1]==True&&PrimeQ[n-(1+Mod[n, 2])Prime[k]]==True, 1, 0], {k, 1, PrimePi[(n-1)/(2+Mod[n, 2])]}]
Do[Print[n, " ", a[n]], {n, 1, 100}]

Crossrefs

Cf. A001359, A006512, A002375, A046927, A219185, A219157, A218867, A219055, A219966.

Sequence in context: A321913 A247378 A094102 * A063746 A293429 A367313

Adjacent sequences: A220088 A220089 A220090 * A220092 A220093 A220094

Keyword

nonn

Author

Zhi-Wei Sun, Dec 04 2012

Status

approved