cp's OEIS Frontend

A237643 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.

%I A237643 #19 Nov 07 2024 08:30:27
%S A237643 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 A237643 88,79,85,113,82,155,114,141,178,132,124,176,155,96,135,176,146,148,
%U A237643 126,125,183,191,185,194,166,261,378,230,278,203,199,161,293,286,175,274
%N A237643 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 A237643 Conjecture: a(n) is always positive. Moreover, a(n) <= 2*prime(n) for all n > 0.
%C A237643 Note that a(15) = 94 = 2*prime(15).
%H A237643 Zhi-Wei Sun, <a href="/A237643/b237643.txt">Table of n, a(n) for n = 1..100</a>
%H A237643 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 A237643 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 A237643 q[m_,n_]:=Length[Union[Table[Mod[PrimePi[k*n],n],{k,1,m}]]]
%t A237643 Do[Do[If[q[m,n]==n,Print[n," ",m];Goto[aa]],{m,n,2*Prime[n]}];
%t A237643 Print[n," ",0];Label[aa];Continue,{n,1,60}]
%Y A237643 Cf. A000720, A237578, A237597, A237598, A237612, A237614, A237656.
%K A237643 nonn
%O A237643 1,2
%A A237643 _Zhi-Wei Sun_, Feb 10 2014