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