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A205386 Least k such that n divides s(k)-s(j) for some j where s(j)=(1/2)C(2j,j).

%I A205386 #7 Dec 25 2023 18:17:07
%S A205386 2,2,3,4,4,6,3,4,3,7,7,6,5,6,7,4,4,9,7,8,6,7,10,6,4,9,6,6,5,7,11,4,7,
%T A205386 4,7,9,9,7,8,8,5,6,15,9,9,10,13,6,7,8
%N A205386 Least k such that n divides s(k)-s(j) for some j<k, where s(j)=(1/2)C(2j,j).
%C A205386 See A204892 for a discussion and guide to related sequences.
%t A205386 s[n_] := s[n] = (1/2) Binomial[2 n, n]; z1 = 700; z2 = 50;
%t A205386 Table[s[n], {n, 1, 30}]    (* A001700 *)
%t A205386 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205386 Table[u[m], {m, 1, z1}]     (* A205384 *)
%t A205386 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205386 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205386 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A205386 Table[d[n], {n, 1, z2}]     (* A205385 *)
%t A205386 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A205386 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A205386 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A205386 Table[k[n], {n, 1, z2}]        (* A205386 *)
%t A205386 Table[j[n], {n, 1, z2}]        (* A205387 *)
%t A205386 Table[s[k[n]], {n, 1, z2}]     (* A205388 *)
%t A205386 Table[s[j[n]], {n, 1, z2}]     (* A205389 *)
%t A205386 Table[s[k[n]] - s[j[n]], {n, 1, z2}]       (* A205390 *)
%t A205386 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]   (* A205391 *)
%Y A205386 Cf. A001700, A204892, A205391.
%K A205386 nonn
%O A205386 1,1
%A A205386 _Clark Kimberling_, Jan 27 2012