A232894 -id:A232894 - cp's OEIS frontend

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

Zhi-Wei Sun, Dec 02 2013

nonn

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.

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

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

Cf. A000984, A232616, A232862, A232894.

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}]

cp's OEIS Frontend

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.

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