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A205720 Numbers k for which 10 divides prime(k)-prime(j) for some j each k occurs once for each such j.

%I A205720 #9 Mar 30 2012 18:58:11
%S A205720 6,7,9,9,10,11,12,12,13,13,14,14,14,15,15,15,16,16,16,16,17,17,18,18,
%T A205720 18,19,19,19,19,20,20,20,20,21,21,21,21,21,22,22,22,23,23,23,23,23,23,
%U A205720 24,24,24,24,25,25,25,25,25,26,26,26,26,26,27,27,27,27,27,27
%N A205720 Numbers k for which 10 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j.
%C A205720 For a guide to related sequences, see A205558.
%e A205720 The first six terms match these differences:
%e A205720 p(6)-p(2)=13-3=10=10*1
%e A205720 p(7)-p(4)=17-7=10=10*1
%e A205720 p(9)-p(2)=23-3=20=10*2
%e A205720 p(9)-p(6)=23-13=10=10*1
%e A205720 p(10)-p(8)=29-19=10=10*1
%e A205720 p(11)-p(5)=31-11=20=10*2
%t A205720 s[n_] := s[n] = Prime[n]; z1 = 900; z2 = 70;
%t A205720 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205720 Table[s[n], {n, 1, 30}]        (* A000040 *)
%t A205720 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205720 Table[u[m], {m, 1, z1}]        (* A204890 *)
%t A205720 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205720 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205720 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205720 c = 10; t = d[c]               (* A205718 *)
%t A205720 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205720 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205720 Table[k[n], {n, 1, z2}]        (* A205720 *)
%t A205720 Table[j[n], {n, 1, z2}]        (* A205721 *)
%t A205720 Table[s[k[n]], {n, 1, z2}]     (* A205722 *)
%t A205720 Table[s[j[n]], {n, 1, z2}]     (* A205723 *)
%t A205720 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205724 *)
%t A205720 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205725 *)
%Y A205720 Cf. A205558, A204892, A204890, A205725.
%K A205720 nonn
%O A205720 1,1
%A A205720 _Clark Kimberling_, Jan 31 2012