A239428 - OEIS

%I #19 Mar 20 2014 13:41:25

%S 0,0,0,0,0,0,0,1,0,1,1,2,2,2,3,2,3,3,2,4,2,4,2,2,3,1,4,2,3,3,2,4,1,5,

%T 4,5,6,4,6,4,5,4,3,5,2,4,2,2,2,1,2,1,2,3,3,2,3,3,5,6,4,6,5,7,4,5,5,4,

%U 5,3,5,6,5,6,4,6,4,6,5,6

%N Number of ordered ways to write n = k + m with 0 < k <= m such that pi(2*k) - pi(k) and pi(2*m) - pi(m) are both prime, where pi(x) denotes the number of primes not exceeding x.

%C Conjecture: a(n) > 0 for all n > 9, and a(n) = 1 only for n = 8, 10, 11, 26, 33, 50, 52.

%C This implies that there are infinitely many positive integers n with pi(2*n) - pi(n) prime.

%C Recall that Bertrand's postulate proved by Chebyshev in 1850 asserts that pi(2*n) - pi(n) > 0 for all n > 0.

%H Zhi-Wei Sun, <a href="/A239428/b239428.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.

%e a(11) = 1 since 11 = 4 + 7 with pi(2*4) - pi(4) = 4 - 2 = 2 and pi(2*7) - pi(7) = 6 - 4 = 2 both prime.

%e a(26) = 1 since 26 = 13 + 13 with pi(2*13) - pi(13) = 9 - 6 = 3 prime.

%e a(33) = 1 since 33 = 6 + 27 with pi(2*6) - pi(6) = 5 - 3 = 2 and pi(2*27) - pi(27) = 16 - 9 = 7 both prime.

%e a(50) = 1 since 50 = 23 + 27 with pi(2*23) - pi(23) = 14 - 9 = 5 and pi(2*27) - pi(27) = 16 - 9 = 7 both prime.

%e a(52) = 1 since 52 = 21 + 31 with pi(2*21) - pi(21) = 13 - 8 = 5 and pi(2*31) - pi(31) = 18 - 11 = 7 both prime.

%t p[n_]:=PrimeQ[PrimePi[2n]-PrimePi[n]]

%t a[n_]:=Sum[If[p[k]&&p[n-k],1,0],{k,1,n/2}]

%t Table[a[n],{n,1,80}]

%Y Cf. A000040, A000720, A108954.

%K nonn

%O 1,12

%A _Zhi-Wei Sun_, Mar 20 2014