cp's OEIS Frontend

A237705 Number of primes p < n with pi(n-p) prime, where pi(.) is given by A000720.

%I A237705 #8 Feb 12 2014 03:15:09
%S A237705 0,0,0,0,1,2,2,3,2,2,2,1,2,3,2,3,3,2,3,3,2,4,4,3,3,1,1,3,3,2,2,1,2,6,
%T A237705 6,5,5,4,3,5,5,4,5,5,4,6,6,6,6,3,3,5,5,5,5,2,2,5,5,3,4,5,4,8,8,3,3,1,
%U A237705 2,8
%N A237705 Number of primes p < n with pi(n-p) prime, where pi(.) is given by A000720.
%C A237705 Conjecture: (i) a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 12, 26, 27, 32, 68.
%C A237705 (ii) For any integer n > 5, there is a prime p <= n with pi(n+p) prime.
%C A237705 (iii) If n > 32, then pi((n-p)^2) is prime for some prime p < n. Also, for each n > 6 there is an odd prime p < 2*n with pi((n - (p-1)/2)^2) prime.
%C A237705 (iv) Any integer n > 11 can be written as p + q with p and pi(q^2 + q + 1) both prime.
%C A237705 (v) Each integer n > 34 can be written as k + m with k and m positive integers such that pi(k^2) and pi(2*m^2) are both prime.
%H A237705 Zhi-Wei Sun, <a href="/A237705/b237705.txt">Table of n, a(n) for n = 1..10000</a>
%e A237705 a(5) = 1 since 2 and pi(5-2) = pi(3) = 2 are both prime.
%e A237705 a(12) = 1 since 7 and pi(12-7) = pi(5) = 3 are both prime.
%e A237705 a(15) = 2 since 3 and pi(15-3) = pi(12) = 5 are both prime, and 11 and pi(15-11) = pi(4) = 2 are both prime.
%e A237705 a(26) = 1 since 23 and pi(26-23) = 2 are both prime.
%e A237705 a(27) = 1 since 23 and pi(27-23) = 2 are both prime.
%e A237705 a(32) = 1 since 29 and pi(32-29) = 2 are both prime.
%e A237705 a(68) = 1 since 37 and pi(68-37) = pi(31) = 11 are both prime.
%t A237705 q[n_]:=PrimeQ[PrimePi[n]]
%t A237705 a[n_]:=Sum[If[q[n-Prime[k]],1,0],{k,1,PrimePi[n-1]}]
%t A237705 Table[a[n],{n,1,70}]
%Y A237705 Cf. A000040, A000720, A237284, A237578, A237582.
%K A237705 nonn
%O A237705 1,6
%A A237705 _Zhi-Wei Sun_, Feb 11 2014