A238568 - OEIS

%I #15 May 09 2019 13:40:55

%S 0,1,1,1,2,2,2,1,2,1,3,2,4,3,4,2,2,5,5,3,4,4,8,1,3,3,4,3,4,3,6,3,4,4,

%T 3,4,6,3,5,2,1,8,3,10,6,5,5,9,7,6,3,8,7,9,2,5,5,2,2,9,7,3,5,8,7,6,8,7,

%U 9,9,6,3,7,8,14,5,9,10,8,11

%N a(n) = |{0 < k < n: n^2 - pi(k*n) is prime}|, where pi(x) denotes the number of primes not exceeding x.

%C Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 8, 10, 24, 41.

%C (ii) For any integer n > 6, there is a positive integer k < n with n^2 + pi(k*n) - 1 prime.

%C (iii) If n > 2, then pi(n^2) - pi(k*n) is prime for some 0 < k < n. If n > 1, then pi(n^2) + pi(k*n) - 1 is prime for some 0 < k < n.

%H Zhi-Wei Sun, <a href="/A238568/b238568.txt">Table of n, a(n) for n = 1..4000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014-2016.

%e a(2) = 1 since 2^2 - pi(1*2) = 4 - 1 = 3 is prime.

%e a(3) = 1 since 3^2 - pi(1*3) = 9 - 2 = 7 is prime.

%e a(4) = 1 since 4^2 - pi(3*4) = 16 - 5 = 11 is prime.

%e a(8) = 1 since 8^2 - pi(4*8) = 64 - 11 = 53 is prime.

%e a(10) = 1 since 10^2 - pi(6*10) = 100 - 17 = 83 is prime.

%e a(24) = 1 since 24^2 - pi(14*24) = 576 - 67 = 509 is prime.

%e a(41) = 1 since 41^2 - pi(10*41) = 1681 - 80 = 1601 is prime.

%t p[k_,n_]:=PrimeQ[n^2-PrimePi[k*n]]

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

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

%Y Cf. A000040, A000720, A237578, A237615, A237712, A238570.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Feb 28 2014