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A227920
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Number of ways to write n = x + y + z with y and z distinct and greater than x such that 6*x-1, 6*y-1, 6*x*y-1 are Sophie Germain primes and {6*x-1, 6*x+1}, {6*z-1, 6*z+1}, {6*x*z-1, 6*x*z+1} are twin prime pairs.
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2
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0, 0, 0, 0, 0, 1, 1, 3, 1, 3, 1, 2, 4, 1, 3, 1, 3, 4, 1, 4, 2, 5, 4, 1, 4, 4, 3, 5, 1, 3, 2, 3, 8, 2, 6, 4, 4, 7, 2, 6, 5, 3, 8, 2, 6, 6, 3, 10, 2, 8, 4, 4, 10, 2, 9, 4, 4, 6, 1, 7, 4, 4, 8, 5, 3, 6, 4, 7, 1, 3, 5, 2, 10, 3, 7, 5, 3, 11, 3, 9, 4, 5, 6, 1, 7, 5, 5, 9, 4, 6, 4, 6, 9, 2, 5, 4, 3, 5, 2, 6
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OFFSET
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1,8
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COMMENTS
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By part (i) of the conjecture in the comments in A227923, for any integer n > 5 not equal to 14 we have a(n) > 0, because there are distinct positive integers x, y, z with x = 1 such that 6*x-1, 6*y-1, 6*x*y-1 are Sophie Germain primes and {6*x-1, 6*x+1}, {6*z-1, 6*z+1}, {6*x*z-1, 6*x*z+1} are twin prime pairs.
Conjecture: Any integer n > 2 can be written as x + y + z (x, y, z > 0) such that 6*x-1, 6*y-1, 6x*y-1, 6*z-1 are Sophie Germain primes, and {6*x-1, 6*x+1}, and {6*y-1, 6*y+1} are twin prime pairs.
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LINKS
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EXAMPLE
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a(14) = 1 since 14 = 2 + 7 + 5, and 6*2-1 = 11, 6*7-1 = 41, 6*2*7-1 = 83 are Sophie Germain primes, and {6*2-1, 6*2+1} ={11, 13}, {6*5-1, 6*5+1} = {29, 31}, {6*2*5-1, 6*2*5+1} = {59, 61} are twin prime pairs.
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MATHEMATICA
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SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[12n-1]
TQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]
RQ[n_]:=TQ[n]&&PrimeQ[12n-1]
a[n_]:=Sum[If[RQ[i]&&SQ[j]&&SQ[i*j]&&TQ[n-i-j]&&TQ[i(n-i-j)]&&Abs[n-i-2j]>0, 1, 0], {i, 1, n/3-1}, {j, i+1, n-1-2i}]
Table[a[n], {n, 1, 100}]
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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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