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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.
2
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 02 2013
STATUS
approved