A238224 - OEIS

%I #15 Apr 06 2014 22:29:04

%S 0,1,1,2,2,2,1,3,5,5,3,3,8,4,3,5,2,1,8,2,2,5,3,4,3,6,4,6,7,6,6,4,8,2,

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

%U 6,5,4,7,7,7,3,9,4,5,7,9

%N Number of pairs {j, k} with 0 < j < k <= n and k == 1 (mod j) such that pi(j*n) divides pi(k*n), where pi(.) is given by A000720.

%C Conjecture: a(n) > 0 for all n > 1.

%C This is a refinement of part (i) of the conjecture in A238165.

%C We have verified the conjecture for n up to 21500.

%H Zhi-Wei Sun, <a href="/A238224/b238224.txt">Table of n, a(n) for n = 1..2900</a>

%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014

%e a(18) = 1 since 6 == 1 (mod 1), and pi(1*18) = 7 divides pi(6*18) = 28.

%e a(50) = 1 since 7 == 1 (mod 3), and pi(3*50) = 35 divides pi(7*50) = 70.

%e a(379) = 1 since 353 == 1 (mod 4), and pi(4*379) = 240 divides pi(353*379) = 12480.

%t m[k_,j_,n_]:=Mod[PrimePi[k*n],PrimePi[j*n]]==0

%t a[n_]:=Sum[If[m[j*q+1,j,n],1,0],{j,1,n-1},{q,1,(n-1)/j}]

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

%Y Cf. A000720, A237578, A237597, A237598, A238165.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Feb 20 2014