cp's OEIS Frontend

A238504 Number of primes p <= n with pi(pi((p-1)*n)) prime, where pi(x) denotes the number of primes not exceeding x.

%I A238504 #34 Nov 07 2024 08:30:20
%S A238504 0,0,1,1,2,2,2,3,2,3,2,1,3,2,4,3,1,2,3,1,1,2,1,2,2,1,1,4,1,2,4,4,5,5,
%T A238504 3,4,4,6,4,2,1,1,3,4,6,4,4,1,3,3,2,4,6,2,2,3,3,3,4,10,5,6,7,5,4,5,5,3,
%U A238504 5,6,6,5,5,4,5,6,6,4,3,4
%N A238504 Number of primes p <= n with pi(pi((p-1)*n)) prime, where pi(x) denotes the number of primes not exceeding x.
%C A238504 Conjecture: (i) a(n) > 0 for all n > 2.
%C A238504 (ii) If n > 92, then pi(p*n) is prime for some prime p <= n. If n > 39, then pi(pi(p*n)) is prime for some p <= n.
%C A238504 See also A238902 for another conjecture involving pi(pi(x)).
%H A238504 Zhi-Wei Sun, <a href="/A238504/b238504.txt">Table of n, a(n) for n = 1..5000</a>
%H A238504 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 A238504 a(3) = 1 since 3 and pi(pi((3-1)*3)) = pi(pi(6)) = pi(3) = 2 are both prime.
%e A238504 a(20) = 1 since 3 and pi(pi((3-1)*20)) = pi(pi(40)) = pi(12) = 5 are both prime.
%e A238504 a(48) = 1 since 29 and pi(pi((29-1)*48)) = pi(pi(1344)) = pi(217) = 47 are both prime.
%t A238504 p[k_,n_]:=PrimeQ[PrimePi[PrimePi[(Prime[k]-1)n]]]
%t A238504 a[n_]:=Sum[If[p[k,n],1,0],{k,1,PrimePi[n]}]
%t A238504 Table[a[n],{n,1,80}]
%Y A238504 Cf. A000040, A000720, A237578, A238890, A238902.
%K A238504 nonn
%O A238504 1,5
%A A238504 _Zhi-Wei Sun_, Mar 06 2014