A237656 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.

1, 5, 3, 6, 8, 10, 18, 17, 30, 41, 28, 43, 29, 33, 43, 27, 66, 47, 98, 105, 155, 114, 113, 100, 49, 62, 118, 146, 85, 125, 80, 117, 74, 101, 167, 137, 168, 282, 176, 287, 129, 178, 151, 140, 163, 139, 262, 267, 277, 234, 285, 188, 203, 163, 192, 239, 188, 241, 252, 252

Offset: 1

Author

Zhi-Wei Sun, Feb 10 2014

Keywords

• nonn

Comments

Conjecture: a(n) is always positive. Moreover, a(n) < 2*prime(n+1) - 2 for all n > 0.

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

Examples

			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.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1011 (n = 1..100 from Zhi-Wei Sun)
• Zhi-Wei Sun, On a^n+bn modulo m, preprint, arXiv:1312.1166 [math.NT], 2013-2014.

Crossrefs

Cf. A000290, A000720, A038107, A237643.

Programs

Mathematica

q[m_,n_]:=Length[Union[Table[Mod[PrimePi[k^2],n],{k,1,m}]]]
Do[Do[If[q[m,n]==n,Print[n," ",m];Goto[aa]],{m,n,2*Prime[n+1]-3}];
Print[n," ",0];Label[aa];Continue,{n,1,60}]