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

Author

Zhi-Wei Sun, Feb 26 2014

Keywords

• nonn

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.

Examples

			 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.
		

Links

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

Crossrefs

Cf. A000040, A000720, A001359, A006512.

Programs

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}]