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.

1, 2, 3, 8, 8, 12, 13, 14, 27, 25, 32, 25, 16, 23, 94, 41, 46, 67, 38, 60, 77, 55, 84, 46, 88, 79, 85, 113, 82, 155, 114, 141, 178, 132, 124, 176, 155, 96, 135, 176, 146, 148, 126, 125, 183, 191, 185, 194, 166, 261, 378, 230, 278, 203, 199, 161, 293, 286, 175, 274

Offset

1,2

Comments

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

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

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..100

Zhi-Wei Sun, On a^n+bn modulo m, preprint, arXiv:1312.1166 [math.NT], 2013-2014.

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A100836 A173162 A198104 * A225474 A368235 A329432

Adjacent sequences: A237640 A237641 A237642 * A237644 A237645 A237646

Keyword

nonn

Author

Zhi-Wei Sun, Feb 10 2014

Status

approved