A237497 a(n) = |{0 < k <= n/2: pi(k*(n-k)) is prime}|, where pi(.) is given by A000720.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 08 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 10, and a(n) = 1 for no n > 51. Moreover, for any integer n > 10, there is a positive integer k < n with 2*k + 1 and pi(k*(n-k)) both prime.

Examples

			a(6) = 1 since 6 = 1 + 5 with pi(1*5) = 3 prime.
a(8) = 1 since 8 = 2 + 6 with pi(2*6) = pi(12) = 5 prime.
a(25) = 1 since 25 = 4 + 21 with pi(4*21) = pi(84) = 23 prime.
a(51) = 1 since 51 = 14 + 37 with pi(14*37) = pi(518) = 97 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A000720, A237284, A237291, A237453, A237496.

Programs

Mathematica

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