A238386 a(n) = |{0 < k < n-1: p = prime(k) + pi(n-k) and p + 2 are both prime}|, where pi(.) is given by A000720.

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

Offset

1,6

Comments

Conjecture: (i) a(n) > 0 for all n > 10.

(ii) For any integer n > 4, there is a positive integer k < n such that prime(k)^2 + pi(n-k)^2 is prime.

We have verified part (i) of the conjecture for n up to 10^7.

Links

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

Example

 a(7) = 1 since prime(1) + pi(7-1) = 2 + 3 = 5 and 5 + 2 = 7 are both prime.
a(30) = 1 since prime(16) + pi(30-16) = 53 + 6 = 59 and 59 + 2 are both prime.
a(108) = 1 since prime(15) + pi(108-15) = 47 + 24 = 71 and 71 + 2 = 73 are both prime.

Mathematica

tq[n_]:=PrimeQ[n]&&PrimeQ[n+2]
a[n_]:=Sum[If[tq[Prime[k]+PrimePi[n-k]], 1, 0], {k, 1, n-2}]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000040, A000720, A001359, A006512.

Sequence in context: A301367 A334090 A179195 * A064662 A352193 A024944

Adjacent sequences: A238383 A238384 A238385 * A238387 A238388 A238389

Keyword

nonn

Author

Zhi-Wei Sun, Feb 26 2014

Status

approved