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A219966 Number of ways to write n=p+q+(n mod 2)q with q<=n/2 and p, q, q+6 all prime 2
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 3, 2, 2, 3, 2, 3, 3, 2, 1, 4, 3, 1, 4, 3, 1, 4, 2, 3, 3, 2, 2, 4, 3, 2, 4, 2, 2, 5, 3, 4, 5, 2, 1, 5, 3, 2, 4, 1, 1, 5, 4, 4, 4, 3, 2, 5, 3, 2, 4, 3, 4, 5, 3, 4, 6, 3, 3, 6, 3, 3, 8, 5, 2, 6, 3, 4, 6, 2, 2, 9, 5, 3, 5, 4, 2, 6, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,17

COMMENTS

Conjecture: a(n)>0 for all n>11.

This conjecture is stronger than Goldbach's conjecture and Lemoine's conjecture. It can be further strengthened; see A219055 and the comments there.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

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

EXAMPLE

a(19)=1 since 19=5+2*7 with 5, 7, 7+6 all prime.

a(20)=1 since 20=13+7 with 13, 7, 7+6 all prime.

MATHEMATICA

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

Do[Print[n, " ", a[n]], {n, 1, 10000}]

CROSSREFS

Cf. A219055, A023201, A002375, A046927.

Sequence in context: A271381 A086247 A116452 * A103958 A321696 A321020

Adjacent sequences:  A219963 A219964 A219965 * A219967 A219968 A219969

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Dec 02 2012

STATUS

approved

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Last modified April 24 22:26 EDT 2019. Contains 322446 sequences. (Running on oeis4.)