The On-Line Encyclopedia of Integer Sequences (OEIS)

A235330 revision #6

A235330

Number of ways to write 2*n = p + q with p, q, prime(p) - p + 1 and prime(q) + q + 1 all prime.

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

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OFFSET

1,9

COMMENTS

Conjecture: (i) a(n) > 0 for all n >= 2480.

(ii) If n > 4368 then 2*n+1 can be written as 2*p + q with p and q terms of the sequence A234695.

Parts (i) and (ii) are stronger than Goldbach's conjecture (A045917) and Lemoine's conjecture (A046927) respectively.

LINKS

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

EXAMPLE

a(8) = 1 since 2*8 = 5 + 11 with 5, 11, prime(5) - 5 + 1 = 7 and prime(11) + 11 + 1 = 43 all prime.

MATHEMATICA