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A205018 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j where s(j)=j*(j+1).

%I A205018 #5 Mar 30 2012 18:58:10
%S A205018 2,2,3,2,3,3,4,4,4,3,6,5,7,4,6,8,9,4,10,6,8,6,12,5,7,7,7,5,15,6,16,16,
%T A205018 8,9,8,6,19,10,9,6,21,8,22,7,10,12,24,9,10,7,11,8,27,7,13,11,12,15,30,
%U A205018 8
%N A205018 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j<k, where s(j)=j*(j+1).
%C A205018 See A204892 for a discussion and guide to related sequences.
%t A205018 s[n_] := s[n] = n*(n + 1); z1 = 500; z2 = 60;
%t A205018 Table[s[n], {n, 1, 30}]   (* A002378 *)
%t A205018 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205018 Table[u[m], {m, 1, z1}]   (* A205016 *)
%t A205018 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205018 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205018 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A205018 Table[d[n], {n, 1, z2}]   (* A205017 *)
%t A205018 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A205018 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A205018 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A205018 Table[k[n], {n, 1, z2}]      (* A205018 *)
%t A205018 Table[j[n], {n, 1, z2}]      (* A205028 *)
%t A205018 Table[s[k[n]], {n, 1, z2}]   (* A205029 *)
%t A205018 Table[s[j[n]], {n, 1, z2}]   (* A205030 *)
%t A205018 Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205031 *)
%t A205018 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]  (* A205032 *)
%Y A205018 Cf. A002378, A204892, A205016.
%K A205018 nonn
%O A205018 1,1
%A A205018 _Clark Kimberling_, Jan 22 2012