cp's OEIS Frontend

A263321 Least positive integer m such that the numbers phi(k)*pi(k^2) (k = 1..n) are pairwise incongruent modulo m.

%I A263321 #8 Oct 15 2015 07:20:30
%S A263321 1,3,5,7,11,13,16,19,19,19,29,29,29,37,37,59,59,59,59,59,59,59,59,101,
%T A263321 101,101,133,133,133,133,173,173,173,173,173,173,173,173,173,173,173,
%U A263321 173,175,245,269,269,269,269,379,379
%N A263321 Least positive integer m such that the numbers phi(k)*pi(k^2) (k = 1..n) are pairwise incongruent modulo m.
%C A263321 Part (i) of the conjecture in A263319 implies that a(n) exists for any n > 0.
%C A263321 Conjecture: a(n) <= n^2 for all n > 0, and the only even term is a(7) = 16.
%H A263321 Zhi-Wei Sun, <a href="/A263321/b263321.txt">Table of n, a(n) for n = 1..4000</a>
%H A263321 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory 133(2013), 2794-2812.
%e A263321 a(7) = 16 since the 7 numbers phi(1)*pi(1^2) = 0, phi(2)*pi(2^2) = 2, phi(3)*pi(3^2) = 8, phi(4)*pi(4^2) = 12, phi(5)*pi(5^2) = 36, phi(6)*pi(6^2) = 22 and phi(7)*pi(7^2) = 90 are pairwise incongruent modulo 16, but not so modulo any positive integer smaller than 16.
%t A263321 f[n_]:=f[n]=EulerPhi[n]*PrimePi[n^2]
%t A263321 Le[n_,m_]:=Le[m,n]=Length[Union[Table[Mod[f[k],m],{k,1,n}]]]
%t A263321 Do[n=1;m=1;Label[aa];If[Le[n,m]==n,Goto[bb],m=m+1;Goto[aa]];
%t A263321 Label[bb];Print[n," ",m];If[n<50,n=n+1;Goto[aa]]]
%Y A263321 Cf. A000010, A000290, A000720, A038107, A263317, A263319.
%K A263321 nonn
%O A263321 1,2
%A A263321 _Zhi-Wei Sun_, Oct 14 2015