A232898 Least positive integer m such that {C(2k,k) + k: k = 1,...,m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

1, 2, 7, 5, 10, 12, 9, 24, 31, 22, 59, 25, 27, 30, 42, 56, 123, 66, 57, 72, 84, 78, 73, 132, 136, 57, 99, 80, 129, 211, 170, 226, 121, 170, 126, 129, 238, 218, 157, 132, 348, 198, 388, 103, 171, 166, 247, 181, 205, 352, 194, 136, 430, 226, 117, 224, 237, 292, 364, 241

Offset

1,2

Comments

Conjecture: (i) Let n be any positive integer. Then 0 < a(n) <= n^2/2 + 3. Also, {C(2k,k) - k: k = 1, ..., [n^2/2] + 15} contains a complete system of residues modulo n, where [.] is the floor function.

(ii) For any integer n > 2, neither C(2n,n) + n nor C(2n,n) - n has the form x^m with m > 1.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..250 from Zhi-Wei Sun)

Example

a(2) = 2 since C(2*1,1) + 1 = 3 is odd and C(2*2,2) + 2 = 8 is even.

Mathematica

L[m_, n_]:=Length[Union[Table[Mod[Binomial[2k, k]+k, n], {k, 1, m}]]]
Do[Do[If[L[m, n]==n, Print[n, " ", m]; Goto[aa]], {m, 1, n^2/2+3}];
Print[n, " ", counterexample]; Label[aa]; Continue, {n, 1, 60}]

Crossrefs

Cf. A000984, A232616, A232862, A232894.

Sequence in context: A011051 A024691 A220606 * A319903 A135076 A155546

Adjacent sequences: A232895 A232896 A232897 * A232899 A232900 A232901

Keyword

nonn

Author

Zhi-Wei Sun, Dec 02 2013

Status

approved