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

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

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

Cf. A000040, A045917, A046927, A234695, A235187, A235189.

Sequence in context: A215401 A254606 A175358 * A029394 A357914 A035467

Adjacent sequences: A235327 A235328 A235329 * A235331 A235332 A235333

Keyword

nonn

Author

Zhi-Wei Sun, Jan 05 2014

Status

approved