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A205691 Numbers k for which 6 divides prime(k)-prime(j) for some j each k occurs once for each such j.

%I A205691 #8 Mar 30 2012 18:58:11
%S A205691 5,6,7,7,8,8,9,9,9,10,10,10,10,11,11,11,12,12,12,12,13,13,13,13,13,14,
%T A205691 14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,16,17,17,17,17,17,17,
%U A205691 17,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20
%N A205691 Numbers k for which 6 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j.
%C A205691 For a guide to related sequences, see A205558.
%e A205691 The first six terms match these differences:
%e A205691 p(5)-p(3)=11-5=6=6*1
%e A205691 p(6)-p(4)=13-7=6=6*1
%e A205691 p(7)-p(3)=17-5=12=6*2
%e A205691 p(7)-p(5)=17-11=6=6*1
%e A205691 p(8)-p(4)=19-7=12=6*2
%e A205691 p(8)-p(6)=19-13=6=6*1
%t A205691 s[n_] := s[n] = Prime[n]; z1 = 400; z2 = 80;
%t A205691 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205691 Table[s[n], {n, 1, 30}]        (* A000040 *)
%t A205691 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205691 Table[u[m], {m, 1, z1}]        (* A204890 *)
%t A205691 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205691 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205691 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205691 c = 6; t = d[c]                (* A205690 *)
%t A205691 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205691 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205691 Table[k[n], {n, 1, z2}]        (* A205691 *)
%t A205691 Table[j[n], {n, 1, z2}]        (* A205692 *)
%t A205691 Table[s[k[n]], {n, 1, z2}]     (* A205693 *)
%t A205691 Table[s[j[n]], {n, 1, z2}]     (* A205694 *)
%t A205691 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205695 *)
%t A205691 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205696 *)
%Y A205691 Cf. A205558, A204892, A204890, A205695, A205696.
%K A205691 nonn
%O A205691 1,1
%A A205691 _Clark Kimberling_, Jan 31 2012