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A199800 Number of ways to write n = p+q with p, 6q-1 and 6q+1 all prime 2
0, 0, 1, 2, 2, 2, 2, 3, 2, 3, 0, 4, 2, 4, 3, 2, 2, 3, 3, 5, 3, 3, 3, 4, 4, 3, 2, 4, 3, 5, 3, 4, 3, 5, 5, 6, 3, 4, 3, 5, 5, 5, 6, 5, 4, 5, 5, 6, 7, 5, 4, 5, 4, 7, 6, 4, 4, 4, 5, 6, 6, 5, 6, 7, 4, 5, 2, 4, 7, 5, 7, 4, 5, 6, 7, 7, 7, 5, 6, 4, 7, 4, 7, 7, 6, 5, 3, 5, 8, 7, 7, 5, 5, 6, 4, 5, 4, 5, 8, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

This implies the twin prime conjecture, and it has been verified for n up to 10^9.

Zhi-Wei Sun also made some similar conjectures, for example, any integer n>5 can be written as p+q with p, 2q-3 and 2q+3 all prime, and each integer n>4 can be written as p+q with p, 3q-2+(n mod 2) and 3q+2-(n mod 2) all prime.

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(3)=1 since 3=2+1 with 2, 6*1-1 and 6*1+1 all prime.

MATHEMATICA

a[n_]:=a[n]=Sum[If[PrimeQ[n-k]==True&&PrimeQ[6k-1]==True&&PrimeQ[6k+1]==True, 1, 0], {k, 1, n-1}]

Do[Print[n, " ", a[n]], {n, 1, 100}]

CROSSREFS

Cf. A001359, A006512, A220419, A220413, A173587, A220272, A219842, A219864, A219923.

Sequence in context: A233417 A299741 A074589 * A165035 A236531 A217403

Adjacent sequences:  A199797 A199798 A199799 * A199801 A199802 A199803

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Dec 21 2012

STATUS

approved

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Last modified May 22 06:32 EDT 2019. Contains 323478 sequences. (Running on oeis4.)