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A205130 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j where s(j)=2*j^2-j, the j-th hexagonal number.

%I A205130 #9 Apr 09 2015 14:21:03
%S A205130 2,3,3,5,2,5,3,9,3,5,4,6,4,3,5,17,5,7,6,6,6,4,7,11,7,11,4,11,8,5,9,33,
%T A205130 9,14,8,9,10,6,5,13,11,12,12,5,7,7,13,18,9,14,6,12,14,8,11,13,8,23,16,
%U A205130 6
%N A205130 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j<k, where s(j)=2*j^2-j, the j-th hexagonal number.
%C A205130 See A204892 for a discussion and guide to related sequences.
%t A205130 s[n_] := s[n] = 2 n^2 - n; z1 = 500; z2 = 60;
%t A205130 Table[s[n], {n, 1, 30}] (* A000384, hexagonal numbers *)
%t A205130 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205130 Table[u[m], {m, 1, z1}] (* A205128 *)
%t A205130 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205130 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205130 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A205130 Table[d[n], {n, 1, z2}]        (* A205129 *)
%t A205130 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A205130 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A205130 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A205130 Table[k[n], {n, 1, z2}]        (* A205130 *)
%t A205130 Table[j[n], {n, 1, z2}]        (* A205131 *)
%t A205130 Table[s[k[n]], {n, 1, z2}]     (* A205132 *)
%t A205130 Table[s[j[n]], {n, 1, z2}]     (* A205133 *)
%t A205130 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205134 *)
%t A205130 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205135 *)
%Y A205130 Cf. A000384, A204892, A205135.
%K A205130 nonn
%O A205130 1,1
%A A205130 _Clark Kimberling_, Jan 25 2012