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A205677 Numbers k for which 4 divides prime(k)-prime(j) for some j each k occurs once for each such j.

%I A205677 #8 Mar 30 2012 18:58:11
%S A205677 4,5,5,6,7,7,8,8,8,9,9,9,9,10,10,10,11,11,11,11,11,12,12,12,12,13,13,
%T A205677 13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,15,16,16,16,16,16,16,17,
%U A205677 17,17,17,17,17,17,17,18,18,18,18,18,18,18,19,19,19,19,19,19
%N A205677 Numbers k for which 4 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j.
%C A205677 For a guide to related sequences, see A205558.
%e A205677 The first six terms match these differences:
%e A205677 p(4)-p(2)=7-3=4=4*1
%e A205677 p(5)-p(2)=11-3=8=4*2
%e A205677 p(5)-p(4)=11-7=4=4*1
%e A205677 p(6)-p(3)=13-5=8=4*2
%e A205677 p(7)-p(3)=17-5=12=4*3
%e A205677 p(7)-p(6)=17-13=4=4*1
%t A205677 s[n_] := s[n] = Prime[n]; z1 = 200; z2 = 80;
%t A205677 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205677 Table[s[n], {n, 1, 30}]        (* A000040 *)
%t A205677 u[m_] :=  u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205677 Table[u[m], {m, 1, z1}]        (* A204890 *)
%t A205677 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205677 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205677 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205677 c = 4; t = d[c]                (* A205676 *)
%t A205677 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205677 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205677 Table[k[n], {n, 1, z2}]         (* A205677 *)
%t A205677 Table[j[n], {n, 1, z2}]         (* A205678 *)
%t A205677 Table[s[k[n]], {n, 1, z2}]      (* A205679 *)
%t A205677 Table[s[j[n]], {n, 1, z2}]      (* A205680 *)
%t A205677 Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205681 *)
%t A205677 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205682 *)
%Y A205677 Cf. A204892, A205676, A204890, A205682.
%K A205677 nonn
%O A205677 1,1
%A A205677 _Clark Kimberling_, Jan 30 2012