%I #19 Nov 07 2024 08:30:27

%S 1,2,3,8,8,12,13,14,27,25,32,25,16,23,94,41,46,67,38,60,77,55,84,46,

%T 88,79,85,113,82,155,114,141,178,132,124,176,155,96,135,176,146,148,

%U 126,125,183,191,185,194,166,261,378,230,278,203,199,161,293,286,175,274

%N Least positive integer m such that {A000720(k*n): k = 1, ..., m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

%C Conjecture: a(n) is always positive. Moreover, a(n) <= 2*prime(n) for all n > 0.

%C Note that a(15) = 94 = 2*prime(15).

%H Zhi-Wei Sun, <a href="/A237643/b237643.txt">Table of n, a(n) for n = 1..100</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1312.1166">On a^n+bn modulo m</a>, preprint, arXiv:1312.1166 [math.NT], 2013-2014.

%e a(4) = 8 since {A000720(4*k): k = 1, ..., 8} = {2, 4, 5, 6, 8, 9, 9, 11} contains a complete system of residues modulo 4, but {pi(4*k): k = 1, ..., 7} contains no integer congruent to 3 modulo 4.

%t q[m_,n_]:=Length[Union[Table[Mod[PrimePi[k*n],n],{k,1,m}]]]

%t Do[Do[If[q[m,n]==n,Print[n," ",m];Goto[aa]],{m,n,2*Prime[n]}];

%t Print[n," ",0];Label[aa];Continue,{n,1,60}]

%Y Cf. A000720, A237578, A237597, A237598, A237612, A237614, A237656.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Feb 10 2014