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A205862 Numbers k for which 7 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 A205862 #5 Mar 30 2012 18:58:12
%S A205862 5,8,9,9,10,12,13,13,13,14,14,15,16,16,16,17,17,17,17,18,18,19,20,20,
%T A205862 21,21,21,21,21,22,22,22,22,23,23,24,24,24,24,24,25,25,25,25,25,25,26,
%U A205862 26,26,27,28,28,28,29,29,29,29,29,29,29
%N A205862 Numbers k for which 7 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 A205862 For a guide to related sequences, see A205840.
%e A205862 The first six terms match these differences:
%e A205862 s(5)-s(1) = 8-1 = 7 = 7*1
%e A205862 s(8)-s(6) = 34-13 = 21 = 7*3
%e A205862 s(9)-s(6) = 55-13 = 42 = 7*6
%e A205862 s(9)-s(8) = 55-34 = 21 = 7*3
%e A205862 s(10)-s(4) = 89-5 = 84 = 7*12
%e A205862 s(13)-s(6) = 377-13 = 364 =7*52
%t A205862 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; z2 = 60;
%t A205862 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205862 Table[s[n], {n, 1, 30}]
%t A205862 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205862 Table[u[m], {m, 1, z1}]    (* A204922 *)
%t A205862 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205862 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205862 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205862 c = 7; t = d[c]   (* A205861 *)
%t A205862 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205862 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205862 Table[k[n], {n, 1, z2}]    (* A205862 *)
%t A205862 Table[j[n], {n, 1, z2}]    (* A205863 *)
%t A205862 Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205864 *)
%t A205862 Table[(s[k[n]]-s[j[n]])/c, {n,1,z2}] (* A205865 *)
%Y A205862 Cf. A204892, A205863, A205865.
%K A205862 nonn
%O A205862 1,1
%A A205862 _Clark Kimberling_, Feb 02 2012