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A205712 Numbers k for which 9 divides prime(k)-prime(j) for some j each k occurs once for each such j.

%I A205712 #12 Oct 25 2024 04:52:06
%S A205712 5,9,10,10,11,12,13,13,14,15,15,15,16,17,17,17,18,18,19,19,20,20,21,
%T A205712 21,22,22,22,23,23,23,23,24,24,24,25,25,25,25,26,26,26,26,26,27,27,27,
%U A205712 28,28,28,28,29,29,29,30,30,30,30,31,31,31,31,32,32,32,32,32,33
%N A205712 Numbers k for which 9 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j.
%C A205712 For a guide to related sequences, see A205558.
%e A205712 The first six terms match these differences:
%e A205712 p(5)-p(1)=11-2=9=9*1
%e A205712 p(9)-p(3)=23-5=18=9*2
%e A205712 p(10)-p(1)=29-2=27=9*3
%e A205712 p(10)-p(5)=29-11=18=9*2
%e A205712 p(11)-p(6)=31-13=18=9*2
%e A205712 p(12)-p(8)=37-19=18=9*2
%t A205712 s[n_] := s[n] = Prime[n]; z1 = 900; z2 = 70;
%t A205712 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205712 Table[s[n], {n, 1, 30}]        (* A000040 *)
%t A205712 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205712 Table[u[m], {m, 1, z1}]        (* A204890 *)
%t A205712 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205712 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205712 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205712 c = 9; t = d[c]                (* A205711 *)
%t A205712 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205712 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205712 Table[k[n], {n, 1, z2}]        (* A205712 *)
%t A205712 Table[j[n], {n, 1, z2}]        (* A205713 *)
%t A205712 Table[s[k[n]], {n, 1, z2}]     (* A205714 *)
%t A205712 Table[s[j[n]], {n, 1, z2}]     (* A205715 *)
%t A205712 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205716 *)
%t A205712 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205717 *)
%Y A205712 Cf. A205558, A204892, A204890, A205713, A205717.
%K A205712 nonn
%O A205712 1,1
%A A205712 _Clark Kimberling_, Jan 31 2012