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A205010 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j where s(j)=C(2j-2,j-1).

%I A205010 #5 Mar 30 2012 18:58:10
%S A205010 2,3,4,3,3,4,4,5,4,5,8,7,6,4,8,5,5,4,4,8,7,8,5,7,5,6,7,7,6,8,12,5,8,5,
%T A205010 8,10,10,8,9,9,6,7,16,8,10,11,8,7,8,5,7,10,8,7,10,7,8,6,20,8
%N A205010 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j<k, where s(j)=C(2j-2,j-1).
%C A205010 See A204892 for a discussion and guide to related sequences.
%t A205010 s[n_] := s[n] = Binomial[2 (n - 1), n - 1];
%t A205010  z1 = 700; z2 = 60;
%t A205010 Table[s[n], {n, 1, 30}]   (* A000984 *)
%t A205010 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205010 Table[u[m], {m, 1, z1}]   (* A205008 *)
%t A205010 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205010 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205010 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A205010 Table[d[n], {n, 1, z2}]   (* A205009 *)
%t A205010 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A205010 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A205010 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A205010 Table[k[n], {n, 1, z2}]      (* A205010 *)
%t A205010 Table[j[n], {n, 1, z2}]      (* A205011 *)
%t A205010 Table[s[k[n]], {n, 1, z2}]   (* A205012 *)
%t A205010 Table[s[j[n]], {n, 1, z2}]   (* A205013 *)
%t A205010 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205014 *)
%t A205010 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205015 *)
%Y A205010 Cf. A000984, A204892, A205008.
%K A205010 nonn
%O A205010 1,1
%A A205010 _Clark Kimberling_, Jan 22 2012