A237578 - OEIS

%I #17 Apr 03 2020 01:39:20

%S 0,0,2,2,1,3,2,1,2,2,4,4,1,4,2,5,5,6,2,5,4,6,3,7,3,3,7,5,5,5,10,9,3,7,

%T 6,5,12,3,3,9,10,11,12,7,3,5,11,9,7,10,12,9,10,8,12,11,10,17,15,13,14,

%U 18,4,17,10,9,15,11,14,11,23,11,9,13,12,12,12,11,14,16

%N a(n) = |{0 < k < n: pi(k*n) is prime}|, where pi(.) is given by A000720.

%C Conjecture: a(n) > 0 for all n > 2, and a(n) = 1 only for n = 5, 8, 13. Moreover, for each n = 1, 2, 3, ..., there is a positive integer k < 3*sqrt(n) + 3 with pi(k*n) prime.

%C Note that the least positive integer k with pi(k*38) prime is 21 < 3*sqrt(38) + 3 < 21.5.

%H Zhi-Wei Sun, <a href="/A237578/b237578.txt">Table of n, a(n) for n = 1..2500</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.

%H Zhi-Wei Sun and Lilu Zhao, <a href="https://arxiv.org/abs/2004.01080">On the set {pi(kn): k=1,2,3,...}</a>, arXiv:2004.01080 [math.NT], 2020.

%e a(5) = 1 since pi(1*5) = 3 is prime.

%e a(8) = 1 since pi(4*8) = 11 is prime.

%e a(13) = 1 since pi(10*13) = pi(130) = 31 is prime.

%e a(38) = 3 since pi(21*38) = pi(798) = 139, pi(28*38) = pi(1064) = 179 and pi(31*38) = pi(1178) = 193 are all prime.

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

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

%Y Cf. A000040, A000720, A237453, A237496, A237497.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Feb 09 2014