%I #12 Jan 04 2018 03:51:56

%S 1,5,3,6,8,10,18,17,30,41,28,43,29,33,43,27,66,47,98,105,155,114,113,

%T 100,49,62,118,146,85,125,80,117,74,101,167,137,168,282,176,287,129,

%U 178,151,140,163,139,262,267,277,234,285,188,203,163,192,239,188,241,252,252

%N Least positive integer m such that {A000720(k^2): 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+1) - 2 for all n > 0.

%C Note that a(21) = 155 = 2*prime(22) - 3.

%H Chai Wah Wu, <a href="/A237656/b237656.txt">Table of n, a(n) for n = 1..1011</a> (n = 1..100 from Zhi-Wei Sun)

%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(5) = 8 since {A000720(k^2): k = 1, ..., 8} = {0, 2, 4, 6, 9, 11, 15, 18} contains a complete system of residues modulo 5, but {A000720(k^2): k = 1, ..., 7} contains no integer congruent to 3 modulo 5.

%t q[m_,n_]:=Length[Union[Table[Mod[PrimePi[k^2],n],{k,1,m}]]]

%t Do[Do[If[q[m,n]==n,Print[n," ",m];Goto[aa]],{m,n,2*Prime[n+1]-3}];

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

%Y Cf. A000290, A000720, A038107, A237643.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Feb 10 2014