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A205705 Numbers k for which 8 divides prime(k)-prime(j) for some j each k occurs once for each such j.

%I A205705 #8 Mar 30 2012 18:58:11
%S A205705 5,6,8,8,9,10,10,11,11,12,12,12,13,14,14,14,15,15,15,16,16,16,16,17,
%T A205705 17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,22,22,22,22,
%U A205705 22,23,23,23,23,23,23,24,24,24,25,25,25,25,26,26,26,26,26,26
%N A205705 Numbers k for which 8 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j.
%C A205705 For a guide to related sequences, see A205558.
%e A205705 The first six terms match these differences:
%e A205705 p(5)-p(2)=11-3=8=8*1
%e A205705 p(6)-p(3)=13-5=8=8*1
%e A205705 p(8)-p(2)=19-3=16=8*2
%e A205705 p(8)-p(5)=19-11=8=8*1
%e A205705 p(9)-p(4)=23-7=16=8*2
%e A205705 p(10)-p(3)=29-5=24=8*3
%t A205705 s[n_] := s[n] = Prime[n]; z1 = 900; z2 = 70;
%t A205705 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205705 Table[s[n], {n, 1, 30}]     (* A000040 *)
%t A205705 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205705 Table[u[m], {m, 1, z1}]     (* A204890 *)
%t A205705 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205705 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205705 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205705 c = 8; t = d[c]             (* A205704 *)
%t A205705 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205705 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205705 Table[k[n], {n, 1, z2}]         (* A205705 *)
%t A205705 Table[j[n], {n, 1, z2}]         (* A205706 *)
%t A205705 Table[s[k[n]], {n, 1, z2}]      (* A205707 *)
%t A205705 Table[s[j[n]], {n, 1, z2}]      (* A205708 *)
%t A205705 Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205709 *)
%t A205705 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205710 *)
%Y A205705 Cf. A205558, A204892, A204890, A205706, A205710.
%K A205705 nonn
%O A205705 1,1
%A A205705 _Clark Kimberling_, Jan 31 2012