A238504 - OEIS

%I #34 Nov 07 2024 08:30:20

%S 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 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 5,6,6,5,5,4,5,6,6,4,3,4

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

%C Conjecture: (i) a(n) > 0 for all n > 2.

%C (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 See also A238902 for another conjecture involving pi(pi(x)).

%H Zhi-Wei Sun, <a href="/A238504/b238504.txt">Table of n, a(n) for n = 1..5000</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(3) = 1 since 3 and pi(pi((3-1)*3)) = pi(pi(6)) = pi(3) = 2 are both prime.

%e a(20) = 1 since 3 and pi(pi((3-1)*20)) = pi(pi(40)) = pi(12) = 5 are both prime.

%e a(48) = 1 since 29 and pi(pi((29-1)*48)) = pi(pi(1344)) = pi(217) = 47 are both prime.

%t p[k_,n_]:=PrimeQ[PrimePi[PrimePi[(Prime[k]-1)n]]]

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

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

%Y Cf. A000040, A000720, A237578, A238890, A238902.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Mar 06 2014