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A205867 Numbers k for which 8 divides s(k)-s(j) for some j each k occurs once for each such j; s(k) denotes the (k+1)-st Fibonacci number.

%I A205867 #5 Mar 30 2012 18:58:12
%S A205867 6,7,7,8,10,11,12,12,13,13,13,14,14,15,16,16,16,17,17,18,18,18,18,19,
%T A205867 19,19,19,19,20,20,20,21,22,22,22,22,23,23,23,24,24,24,24,24,25,25,25,
%U A205867 25,25,25
%N A205867 Numbers k for which 8 divides s(k)-s(j) for some j<k; each k occurs once for each such j; s(k) denotes the (k+1)-st Fibonacci number.
%C A205867 For a guide to related sequences, see A205840.
%e A205867 The first six terms match these differences:
%e A205867 s(6)-s(4) = 13-5 = 8 = 8*1
%e A205867 s(7)-s(4) = 21-5 = 16 = 8*2
%e A205867 s(7)-s(6) = 21-13 = 8 = 8*1
%e A205867 s(8)-s(2) = 34-2 = 32 = 8*4
%e A205867 s(10)-s(1) = 89-1 = 88 = 8*11
%e A205867 s(11)-s(5) = 144-8 = 136 =8*17
%t A205867 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 600; z2 = 50;
%t A205867 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205867 Table[s[n], {n, 1, 30}]
%t A205867 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205867 Table[u[m], {m, 1, z1}]   (* A204922 *)
%t A205867 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205867 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205867 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205867 c = 8; t = d[c]    (* A205866 *)
%t A205867 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205867 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205867 Table[k[n], {n, 1, z2}]      (* A205867 *)
%t A205867 Table[j[n], {n, 1, z2}]        (* A205868 *)
%t A205867 Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205869 *)
%t A205867 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205870 *)
%Y A205867 Cf. A204892, A205868, A205870.
%K A205867 nonn
%O A205867 1,1
%A A205867 _Clark Kimberling_, Feb 02 2012