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A236566

Number of ordered ways to write 2*n = p + q with p, q and prime(p + 2) + 2 all prime.

0, 0, 1, 2, 2, 1, 2, 3, 2, 1, 3, 2, 1, 2, 1, 1, 4, 2, 1, 2, 3, 3, 4, 5, 4, 4, 5, 2, 4, 4, 3, 5, 3, 1, 5, 6, 4, 3, 6, 2, 4, 8, 4, 3, 6, 3, 4, 3, 3, 4, 5, 4, 3, 6, 6, 5, 8, 3, 4, 7, 2, 3, 5, 2, 4, 4, 3, 3, 6, 5, 4, 6, 3, 4, 7, 3, 5, 4, 2, 4, 4, 1, 2, 7, 4, 2, 5, 3, 5, 6, 4, 4, 4, 2, 3, 4, 4, 4, 5, 2

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OFFSET

1,4

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 2.

(ii) If n > 30, then 2*n + 1 can be written as 2*p + q with p, q and prime(p + 2) + 2 all prime.

Part (i) implies both the Goldbach conjecture and the twin prime conjecture. If all primes p with prime(p + 2) + 2 are smaller than an even number N > 2, then for any such a prime p the number N! + N - p is in the interval (N!, N! + N) and hence not prime.

Similarly, part (ii) implies both Lemoine's conjecture (cf. A046927) and the twin prime conjecture.

We have verified part (i) of the conjecture for n up to 2*10^8.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

EXAMPLE

a(10) = 1 since 2*10 = 3 + 17 with 3, 17 and prime(3 + 2) + 2 = 11 + 2 = 13 all prime.

a(589) = 1 since 2*589 = 577 + 601 with 577, 601 and prime(577 + 2) + 2 = 4229 + 2 = 4231 all prime.

MATHEMATICA

p[m_]:=PrimeQ[Prime[m+2]+2]

a[n_]:=Sum[If[p[Prime[k]]&&PrimeQ[2n-Prime[k]], 1, 0], {k, 1, PrimePi[2n-1]}]

Table[a[n], {n, 1, 100}]

CROSSREFS