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A205698 Numbers k for which 7 divides prime(k)-prime(j) for some j each k occurs once for each such j.

%I A205698 #8 Mar 30 2012 18:58:11
%S A205698 7,8,9,11,11,12,12,13,14,15,15,16,17,17,17,18,18,18,19,19,20,20,21,21,
%T A205698 21,21,22,22,22,23,23,24,24,24,24,25,25,25,26,26,26,26,26,27,27,27,27,
%U A205698 27,28,28,28,28,29,29,29,30,30,30,31,31,31,31,32,32,32,32,32
%N A205698 Numbers k for which 7 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j.
%C A205698 For a guide to related sequences, see A205558.
%e A205698 The first six terms match these differences:
%e A205698 p(7)-p(2)=17-3=14=7*2
%e A205698 p(8)-p(3)=19-5=14=7*2
%e A205698 p(9)-p(1)=23-2=21=7*3
%e A205698 p(11)-p(2)=31-3=28=7*4
%e A205698 p(11)-p(7)=31-17=14=7*2
%e A205698 p(12)-p(1)=37-2=35=7*5
%t A205698 s[n_] := s[n] = Prime[n]; z1 = 1200; z2 = 80;
%t A205698 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205698 Table[s[n], {n, 1, 30}]        (* A000040 *)
%t A205698 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205698 Table[u[m], {m, 1, z1}]        (* A204890 *)
%t A205698 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205698 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205698 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205698 c = 7; t = d[c]                (* A205697 *)
%t A205698 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205698 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205698 Table[k[n], {n, 1, z2}]         (* A205698 *)
%t A205698 Table[j[n], {n, 1, z2}]         (* A205699 *)
%t A205698 Table[s[k[n]], {n, 1, z2}]      (* A205700 *)
%t A205698 Table[s[j[n]], {n, 1, z2}]      (* A205701 *)
%t A205698 Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205702 *)
%t A205698 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205703 *)
%Y A205698 Cf. A205558, A204892, A204890, A205699, A205703.
%K A205698 nonn
%O A205698 1,1
%A A205698 _Clark Kimberling_, Jan 31 2012