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A187759 Number of ways to write n=x+y (0<x<y<n) with 6x-1, 6x+1, 6y-1 and 6y+1 all prime. 4
0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 0, 2, 1, 3, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 1, 2, 3, 2, 6, 1, 3, 1, 2, 4, 3, 4, 4, 1, 3, 1, 3, 5, 2, 6, 1, 3, 2, 2, 5, 2, 5, 2, 3, 1, 2, 3, 5, 2, 4, 0, 0, 3, 1, 6, 2, 3, 3, 1, 5, 1, 5, 3, 3, 3, 1, 4, 2, 3, 3, 0, 3, 3, 3, 4, 1, 3, 1, 2, 3, 2, 4, 2, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Conjecture: If n>200 is not among 211, 226, 541, 701, then a(n)>0.

This essentially follows from the conjecture related to A219157, since n=x+y for some positive integers x and y with 6x-1,6x+1,6y-1,6y+1 all prime if and only if 6n=p+q for some twin prime pairs {p,p-2} and {q,q+2}.

Similarly, the conjecture related to A218867 implies that any integer n>491 can be written as x+y (0<x<=y<n) with 6x+1, 6x+5, 6y+1 and 6y+5 all prime; and the conjecture related to A219055 implies that any integer n>1600 not among 2729 and 4006 can be written as x+y (0<x<=y<n) with 2x-3, 2x+3, 2y-3 and 2y+3 all prime.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000

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

EXAMPLE

a(9)=1 since 9=2+7 with 6*2-1, 6*2+1, 6*7-1 and 6*7+1 all prime.

MATHEMATICA

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

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

PROG

(PARI) a(n)=sum(x=1, (n-1)\2, isprime(6*x-1)&&isprime(6*x+1)&&isprime(6*n-6*x-1)&&isprime(6*n-6*x+1)) \\ Charles R Greathouse IV, Feb 28 2013

CROSSREFS

Cf. A001359, A006512, A219157, A219185, A199920, A187757, A187758, A218867, A219055.

Sequence in context: A242748 A266012 A202111 * A270645 A268059 A037226

Adjacent sequences:  A187756 A187757 A187758 * A187760 A187761 A187762

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 03 2013

STATUS

approved

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Last modified April 7 04:20 EDT 2020. Contains 333292 sequences. (Running on oeis4.)