%I #9 Apr 09 2015 14:21:03

%S 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 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 6

%N 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 See A204892 for a discussion and guide to related sequences.

%t s[n_] := s[n] = 2 n^2 - n; z1 = 500; z2 = 60;

%t Table[s[n], {n, 1, 30}] (* A000384, hexagonal numbers *)

%t u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]

%t Table[u[m], {m, 1, z1}] (* A205128 *)

%t v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]

%t w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]

%t d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]

%t Table[d[n], {n, 1, z2}] (* A205129 *)

%t k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]

%t m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]

%t j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2

%t Table[k[n], {n, 1, z2}] (* A205130 *)

%t Table[j[n], {n, 1, z2}] (* A205131 *)

%t Table[s[k[n]], {n, 1, z2}] (* A205132 *)

%t Table[s[j[n]], {n, 1, z2}] (* A205133 *)

%t Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205134 *)

%t Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205135 *)

%Y Cf. A000384, A204892, A205135.

%K nonn

%O 1,1

%A _Clark Kimberling_, Jan 25 2012