

A235330


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


5



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
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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



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

p[n_] := PrimeQ[n] && PrimeQ[Prime[n]  n + 1];
q[n_] := PrimeQ[n] && PrimeQ[Prime[n] + n + 1];
a[n_] := Sum[If[p[k] && q[2 n  k], 1, 0], {k, 1, 2 n  1}];
Table[a[n], {n, 1, 100}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



