A237496 Number of ordered ways to write n = k + m (0 < k <= m) with pi(k) + pi(m) - 2 prime, where pi(.) is given by A000720.

0, 0, 0, 0, 0, 1, 2, 4, 4, 3, 2, 3, 3, 3, 5, 3, 1, 4, 5, 5, 7, 4, 1, 2, 1, 1, 1, 1, 1, 3, 6, 7, 8, 8, 8, 8, 8, 9, 11, 11, 11, 11, 9, 7, 7, 4, 1, 2, 1, 2, 3, 5, 7, 10, 14, 14, 14, 10, 6, 10, 14, 16, 19, 16, 13, 12, 11, 10, 7, 6, 5, 3, 3, 4, 3, 6, 9, 13, 17, 18

Offset: 1

Author

Zhi-Wei Sun, Feb 08 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) Any integer n > 23 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) prime. Also, each integer n > 25 can be written as k + m (k > 0 and m > 0) with pi(k) + pi(m) - 1 prime.

Examples

			a(6) = 1 since 6 = 3 + 3 with pi(3) + pi(3) - 2 = 2 + 2 - 2 = 2 prime.
a(17) = 1 since 17 = 2 + 15 with pi(2) + pi(15) - 2 = 1 + 6 - 2 = 5 prime.
a(99) = 1 since 99 = 1 + 98 with pi(1) + pi(98) - 2 = 0 + 25 - 2 = 23 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A000040, A000720, A232465, A237284, A237291, A237453, A237497.

Programs

Mathematica

PQ[n_]:=n>0&&PrimeQ[n]
p[k_,m_]:=PQ[PrimePi[k]+PrimePi[m]-2]
a[n_]:=Sum[If[p[k,n-k],1,0],{k,1,n/2}]
Table[a[n],{n,1,80}]