cp's OEIS Frontend

A237578 a(n) = |{0 < k < n: pi(k*n) is prime}|, where pi(.) is given by A000720.

%I A237578 #17 Apr 03 2020 01:39:20
%S A237578 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 A237578 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 A237578 18,4,17,10,9,15,11,14,11,23,11,9,13,12,12,12,11,14,16
%N A237578 a(n) = |{0 < k < n: pi(k*n) is prime}|, where pi(.) is given by A000720.
%C A237578 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 A237578 Note that the least positive integer k with pi(k*38) prime is 21 < 3*sqrt(38) + 3 < 21.5.
%H A237578 Zhi-Wei Sun, <a href="/A237578/b237578.txt">Table of n, a(n) for n = 1..2500</a>
%H A237578 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 A237578 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 A237578 a(5) = 1 since pi(1*5) = 3 is prime.
%e A237578 a(8) = 1 since pi(4*8) = 11 is prime.
%e A237578 a(13) = 1 since pi(10*13) = pi(130) = 31 is prime.
%e A237578 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 A237578 a[n_]:=Sum[If[PrimeQ[PrimePi[k*n]],1,0],{k,1,n-1}]
%t A237578 Table[a[n],{n,1,80}]
%Y A237578 Cf. A000040, A000720, A237453, A237496, A237497.
%K A237578 nonn
%O A237578 1,3
%A A237578 _Zhi-Wei Sun_, Feb 09 2014