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A205402 Least k such that n divides s(k)-s(j) for some j < k, where s(j) = floor((j+1)^2/2)/2 (quarter-squares).

%I A205402 #19 Dec 07 2021 13:08:04
%S A205402 2,3,3,4,4,6,5,5,8,6,6,7,9,7,7,8,11,8,8,11,9,12,9,9,14,10,11,10,10,11,
%T A205402 14,11,12,11,11,12,13,12,15,12,12,16,13,14,13,20,13,13,19,14,17,14,20,
%U A205402 14,14,16,21,15,25,15,17,15,15,19,17,16,28,16,17,16,16,17,26
%N A205402 Least k such that n divides s(k)-s(j) for some j < k, where s(j) = floor((j+1)^2/2)/2 (quarter-squares).
%C A205402 See A204892 for a discussion and guide to related sequences.
%t A205402 s[n_] := s[n] = (1/2) Floor[(n + 1)^2/2];
%t A205402 z1 = 1000; z2 = 80;
%t A205402 Table[s[n], {n, 1, 30}]   (* A002620, quarter-squares *)
%t A205402 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205402 Table[u[m], {m, 1, z1}]   (* A205400 *)
%t A205402 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205402 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205402 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A205402 Table[d[n], {n, 1, z2}]   (* A205401 *)
%t A205402 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A205402 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A205402 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A205402 Table[k[n], {n, 1, z2}]        (* A205402 *)
%t A205402 Table[j[n], {n, 1, z2}]        (* A205403 *)
%t A205402 Table[s[k[n]], {n, 1, z2}]     (* A205404 *)
%t A205402 Table[s[j[n]], {n, 1, z2}]     (* A205405 *)
%t A205402 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205406 *)
%t A205402 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A198293 *)
%Y A205402 Cf. A002620, A205400, A204892.
%K A205402 nonn
%O A205402 1,1
%A A205402 _Clark Kimberling_, Jan 27 2012
%E A205402 Name edited by _Clark Kimberling_, Dec 06 2021