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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. 9
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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A000040, A001359, A002372, A002375, A006512, A046927, A236531.

Sequence in context: A071694 A072781 A340094 * A046923 A184703 A309287

Adjacent sequences:  A236563 A236564 A236565 * A236567 A236568 A236569

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 28 2014

STATUS

approved

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Last modified December 5 03:03 EST 2021. Contains 349530 sequences. (Running on oeis4.)