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A187759
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Number of ways to write n=x+y (0<x<y<n) with 6x-1, 6x+1, 6y-1 and 6y+1 all prime.
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4
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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
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OFFSET
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1,8
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COMMENTS
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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.
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LINKS
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EXAMPLE
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a(9)=1 since 9=2+7 with 6*2-1, 6*2+1, 6*7-1 and 6*7+1 all prime.
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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