A235187 Number of ordered ways to write 2*n = p + q with p, q and prime(p) + q - 1 all prime.

0, 0, 1, 1, 2, 1, 1, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 3, 1, 2, 5, 5, 2, 5, 3, 2, 5, 2, 1, 6, 2, 4, 4, 1, 5, 3, 4, 3, 6, 6, 3, 5, 5, 2, 9, 3, 3, 7, 2, 4, 7, 6, 3, 6, 7, 5, 4, 4, 4, 12, 3, 2, 5, 3, 3, 9, 3, 1, 7, 4, 2, 8, 6, 3, 8, 3, 4, 7, 6, 3, 10, 3, 3, 10, 8, 3, 11, 5, 3, 10, 6, 1, 9, 8, 2, 7, 4, 3, 9, 4

Offset

1,5

Comments

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

(ii) Any integer n > 4 can be written as p + q with q > 0 such that p and p - 1 + prime(q) are both prime.

(iii) Each integer n > 7 can be written as p + q with q > 0 such that prime(p) + sigma(q) is prime, where sigma(q) denotes the sum of all positive divisors of q.

Clearly, part (i) is stronger than Goldbach's conjecture.

Links

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

Example

 a(7) = 1 since 2*7 = 7 + 7 with 7 and prime(7) + 7 - 1 = 17 + 6 = 23 both prime.
a(14) = 1 since 2*14 = 11 + 17 with 11, 17 and prime(11) + 16 = 47 all prime.
a(92) = 1 since 2*92 = 47 + 137 with 47, 137 and prime(47) + 136 = 347 all prime.

Mathematica

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

Crossrefs

Cf. A000040, A000203, A045917, A234694.

Sequence in context: A156265 A232992 A165125 * A029333 A029261 A359384

Adjacent sequences: A235184 A235185 A235186 * A235188 A235189 A235190

Keyword

nonn

Author

Zhi-Wei Sun, Jan 04 2014

Status

approved