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A205684 Numbers k for which 5 divides prime(k)-prime(j) for some j each k occurs once for each such j.

%I A205684 #8 Mar 30 2012 18:58:11
%S A205684 4,6,7,7,9,9,10,11,12,12,12,13,13,14,14,14,15,15,15,15,16,16,16,16,17,
%T A205684 17,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,21,22,22,22,23,23,
%U A205684 23,23,23,23,24,24,24,24,25,25,25,25,25,25,26,26,26,26,26,27
%N A205684 Numbers k for which 5 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j.
%C A205684 For a guide to related sequences, see A205558.
%e A205684 The first six terms match these differences:
%e A205684 p(4)-p(1)=7-2=5=5*1
%e A205684 p(6)-p(2)=13-3=10=5*2
%e A205684 p(7)-p(1)=17-2=15=5*3
%e A205684 p(7)-p(4)=17-7=10=5*2
%e A205684 p(9)-p(2)=23-3=20=5*4
%e A205684 p(9)-p(6)=23-13=10=5*2
%t A205684 s[n_] := s[n] = Prime[n]; z1 = 400; z2 = 80;
%t A205684 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205684 Table[s[n], {n, 1, 30}]        (* A000040 *)
%t A205684 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205684 Table[u[m], {m, 1, z1}]        (* A204890 *)
%t A205684 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205684 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205684 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205684 c = 5; t = d[c]                (* A205683 *)
%t A205684 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205684 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205684 Table[k[n], {n, 1, z2}]        (* A205684 *)
%t A205684 Table[j[n], {n, 1, z2}]        (* A205685 *)
%t A205684 Table[s[k[n]], {n, 1, z2}]     (* A205686 *)
%t A205684 Table[s[j[n]], {n, 1, z2}]     (* A205687 *)
%t A205684 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205688 *)
%t A205684 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205689 *)
%Y A205684 Cf. A205558, A204892, A204890, A205685, A205689.
%K A205684 nonn
%O A205684 1,1
%A A205684 _Clark Kimberling_, Jan 30 2012