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A237348
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Number of ordered ways to write n = k + m with k > 0 and m > 0 such that prime(k) + 4 and prime(prime(m)) + 4 are both prime.
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4
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0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 3, 1, 2, 1, 1, 1, 2, 3, 1, 2, 2, 1, 2, 3, 3, 3, 5, 4, 2, 4, 1, 5, 1, 5, 1, 4, 4, 3, 3, 3, 1, 5, 4, 4, 3, 5, 3, 5, 6, 3, 3, 4, 3, 4, 5, 1, 5, 3, 3, 3, 5, 4, 2, 8, 1, 2, 5, 6
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OFFSET
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1,12
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COMMENTS
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Conjecture: For each d = 1, 2, 3, ... there is a positive integer N(d) for which any integer n > N(d) can be written as k + m with k > 0 and m > 0 such that prime(k) + 2*d and prime(prime(m)) + 2*d are both prime. In particular, we may take (N(1), N(2), ..., N(10)) = (2, 11, 4, 15, 31, 4, 2, 77, 4, 7).
This extension of the "Super Twin Prime Conjecture" (posed by the author) implies de Polignac's well-known conjecture that any positive even number can be a difference of two primes infinitely often.
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LINKS
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EXAMPLE
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a(7) = 1 since 7 = 6 + 1 with prime(6) + 4 = 13 + 4 = 17 and prime(prime(1)) + 4 = prime(2) + 4 = 7 both prime.
a(114) = 1 since 114 = 78 + 36 with prime(78) + 4 = 397 + 4 = 401 and prime(prime(36)) + 4 = prime(151) + 4 = 877 + 4 = 881 both prime.
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MATHEMATICA
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pq[n_]:=pq[n]=PrimeQ[Prime[n]+4]
PQ[n_]:=PrimeQ[Prime[Prime[n]]+4]
a[n_]:=Sum[If[pq[k]&&PQ[n-k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 80}]
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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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