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A204900 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k) is the k-th odd prime.

%I A204900 #14 May 15 2017 15:03:32
%S A204900 2,2,4,3,5,4,6,4,8,5,9,6,9,6,11,7,11,8,12,8,14,9,15,9,15,9,16,10,17,
%T A204900 11,18,11,19,11,20,12,21,12,22,13,23,14,23,14,24,15,24,15,25,15,27,16,
%U A204900 28,16,29,16,30,17,31,18,30,18,31,18,32,19,32,19,34,20,34,21,34
%N A204900 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k) is the k-th odd prime.
%C A204900 See A204892 for a discussion and guide to related sequences
%t A204900 s[n_] := s[n] = Prime[n + 1]; z1 = 400; z2 = 50;
%t A204900 Table[s[n], {n, 1, 30}]      (* A065091 *)
%t A204900 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A204900 Table[u[m], {m, 1, z1}]      (* A204898 *)
%t A204900 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A204900 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A204900 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A204900 Table[d[n], {n, 1, z2}]      (* A204899 *)
%t A204900 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A204900 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A204900 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A204900 Table[k[n], {n, 1, z2}]      (* A204900 *)
%t A204900 Table[j[n], {n, 1, z2}]      (* A204901 *)
%t A204900 Table[s[k[n]], {n, 1, z2}]   (* A204902 *)
%t A204900 Table[s[j[n]], {n, 1, z2}]   (* A204903 *)
%t A204900 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204904 *)
%t A204900 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A000034 conjectured *)
%Y A204900 Cf. A000040, A065091, A204892.
%K A204900 nonn
%O A204900 1,1
%A A204900 _Clark Kimberling_, Jan 20 2012