cp's OEIS Frontend

A205450 Least k such that n divides s(k) - s(j) for some j where s(j) is the 2j-th Fibonacci number.

%I A205450 #9 Aug 06 2025 09:17:26
%S A205450 2,2,4,4,3,4,3,6,4,4,6,8,4,7,12,9,5,4,10,4,8,7,7,8,13,5,5,9,8,14,16,
%T A205450 15,12,5,11,12,10,10,16,9,6,8,12,16,14,7,5,12,10,14,20,5,14,5,10,9,20,
%U A205450 8,30,32
%N A205450 Least k such that n divides s(k) - s(j) for some j<k, where s(j) is the 2j-th Fibonacci number.
%C A205450 See A204892 for a discussion and guide to related sequences.
%t A205450 s[n_] := s[n] = Fibonacci[2*n]; z1 = 500; z2 = 60;
%t A205450 Table[s[n], {n, 1, 30}]     (* A001906 *)
%t A205450 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205450 Table[u[m], {m, 1, z1}]     (* A205448 *)
%t A205450 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205450 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205450 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A205450 Table[d[n], {n, 1, z2}]     (* A205449 *)
%t A205450 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A205450 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A205450 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A205450 Table[k[n], {n, 1, z2}]         (* A205450 *)
%t A205450 Table[j[n], {n, 1, z2}]         (* A205451 *)
%t A205450 Table[s[k[n]], {n, 1, z2}]      (* A205452 *)
%t A205450 Table[s[j[n]], {n, 1, z2}]      (* A205453 *)
%t A205450 Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205454 *)
%t A205450 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]  (* A205455 *)
%Y A205450 Cf. A001906, A205455, A204892.
%K A205450 nonn
%O A205450 1,1
%A A205450 _Clark Kimberling_, Jan 27 2012