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A205114 Least k such that n divides L(k)-L(j) for some j satisfying 1<=j where L(j) is the j-th Lucas number (A000032).

%I A205114 #12 Mar 24 2015 06:07:55
%S A205114 2,2,3,4,5,4,5,5,7,5,6,8,7,6,6,10,6,7,10,8,10,7,8,9,7,7,16,7,8,10,16,
%T A205114 11,11,11,10,8,13,10,11,8,11,14,8,8,12,8,9,11,11,16,13,13,12,16,12,10,
%U A205114 17,9,14,10
%N A205114 Least k such that n divides L(k)-L(j) for some j satisfying 1<=j<k, where L(j) is the j-th Lucas number (A000032).
%C A205114 See A204892 for a discussion and guide to related sequences.
%t A205114 s[n_] := s[n] = LucasL[n]; z1 = 500; z2 = 60;
%t A205114 Table[s[n], {n, 1, 30}]    (* A000032 *)
%t A205114 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205114 Table[u[m], {m, 1, z1}]    (* A205112 *)
%t A205114 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205114 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205114 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A205114 Table[d[n], {n, 1, z2}]     (* A205113 *)
%t A205114 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A205114 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A205114 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A205114 Table[k[n], {n, 1, z2}]       (* A205114 *)
%t A205114 Table[j[n], {n, 1, z2}]       (* A205115 *)
%t A205114 Table[s[k[n]], {n, 1, z2}]    (* A205116 *)
%t A205114 Table[s[j[n]], {n, 1, z2}]    (* A205117 *)
%t A205114 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205118 *)
%t A205114 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205119 *)
%Y A205114 Cf. A000032, A204892, A205119.
%K A205114 nonn
%O A205114 1,1
%A A205114 _Clark Kimberling_, Jan 22 2012