A235508 Number of ways to write 2*n = p + q with q > 0 such that p, p*(p+1) - prime(p) and prime(q) - q + 1 are all prime.

0, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 4, 1, 2, 5, 4, 4, 2, 3, 3, 6, 1, 3, 5, 4, 3, 4, 3, 2, 4, 4, 3, 5, 3, 4, 4, 3, 3, 4, 4, 5, 4, 1, 2, 6, 1, 3, 4, 3, 5, 6, 1, 2, 4, 2, 4, 2, 1, 3, 7, 3, 3, 6, 4, 5, 6, 2, 3, 7, 3, 5, 4, 6, 4, 8, 3, 5, 2, 6, 4, 6

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 1.

Links

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

Example

a(7) = 1 since 2*7 = 11 + 3 with 11, 11*12 - prime(11) = 101 and prime(3) - 3 + 1 = 3 all prime.
a(19) = 1 since 2*19 = 37 + 1 with 37, 37*38 - prime(37) = 1249 and prime(1) - 1 + 1 = 2 all prime.
a(98) = 1 since 2*98 = 11 + 185 with 11, 11*12 - prime(11) = 101 and prime(185) - 185 + 1 = 919 all prime.

Mathematica

p[k_]:=PrimeQ[Prime[k](Prime[k]+1)-Prime[Prime[k]]]
q[m_]:=PrimeQ[Prime[m]-m+1]
a[n_]:=Sum[If[p[k]&&q[2n-Prime[k]], 1, 0], {k, 1, PrimePi[2n-1]}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000040, A234695, A234851, A235189, A235330, A235661, A235681.

Sequence in context: A098220 A297032 A346953 * A271778 A326702 A133563

Adjacent sequences: A235505 A235506 A235507 * A235509 A235510 A235511

Keyword

nonn

Author

Zhi-Wei Sun, Jan 14 2014

Status

approved