A204905 - OEIS

%I #7 Mar 30 2012 18:58:08

%S 2,3,2,3,3,4,4,3,5,6,6,4,7,8,4,5,9,9,10,6,5,12,12,5,10,14,6,8,15,8,16,

%T 6,7,18,6,9,19,20,8,7,21,10,22,12,7,24,24,7,14,15,10,14,27,12,8,9,11,

%U 30,30,8

%N Least k such that n divides k^2-j^2 for some j satisfying 1<=j<k.

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

%e 1 divides 2^2-1^2, so a(1)=2

%e 2 divides 3^2-1^2, so a(2)=3

%e 3 divides 2^2-a^2, so a(3)=2

%e 4 divides 3^2-a^2, so a(4)=3

%t s[n_] := s[n] = n^2; z1 = 600; z2 = 60;

%t Table[s[n], {n, 1, 30}]     (* A000290 *)

%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}]     (* A120070 *)

%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}]   (* A204994 *)

%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}]       (* A204905 *)

%t Table[j[n], {n, 1, z2}]       (* A204995 *)

%t Table[s[k[n]], {n, 1, z2}]    (* A204996 *)

%t Table[s[j[n]], {n, 1, z2}]    (* A204997 *)

%t Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204998 *)

%t Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204999 *)

%Y Cf. A000290, A204892.

%K nonn

%O 1,1

%A _Clark Kimberling_, Jan 21 2012