A232894 Least positive integer m such that {Catalan(k) - k: k = 1, ..., m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.

1, 5, 4, 11, 16, 13, 31, 27, 18, 22, 34, 52, 45, 45, 31, 112, 57, 73, 113, 99, 64, 77, 114, 215, 134, 106, 89, 99, 127, 209, 161, 239, 135, 178, 96, 207, 185, 172, 157, 231, 174, 195, 309, 115, 274, 309, 386, 239, 200, 336, 188, 199, 181, 181, 116, 311, 229, 290, 663, 239

Offset

1,2

Comments

Conjecture: (i) Let n be any positive integer. Then 0 < a(n) <= n^2/2 + 7. Also, {Catalan(k) + k: k = 1, ..., [n^2/2] + 23} contains a complete system of residues modulo n, where [.] is the floor function.

(ii) For any integer n > 3, neither Catalan(n) - n nor Catalan(n) + n has the form x^m with m > 1 and x > 1.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..200 from Zhi-Wei Sun)

Example

a(2) = 5 since Catalan(k) - k is even for each k = 1, 2, 3, 4, and Catalan(5) - 5 = 37 is odd.

Mathematica

L[m_, n_]:=Length[Union[Table[Mod[CatalanNumber[k]-k, n], {k, 1, m}]]]
Do[Do[If[L[m, n]==n, Print[n, " ", m]; Goto[aa]], {m, 1, n^2/2+7}];
Print[n, " ", counterexample]; Label[aa]; Continue, {n, 1, 60}]

Crossrefs

Cf. A000108, A232616, A232862, A232898.

Sequence in context: A165367 A166549 A324781 * A262902 A215332 A298779

Adjacent sequences: A232891 A232892 A232893 * A232895 A232896 A232897

Keyword

nonn

Author

Zhi-Wei Sun, Dec 02 2013

Status

approved