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).

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, 14, 11, 12, 11, 11, 12, 13, 12, 15, 12, 12, 16, 13, 14, 13, 20, 13, 13, 19, 14, 17, 14, 20, 14, 14, 16, 21, 15, 25, 15, 17, 15, 15, 19, 17, 16, 28, 16, 17, 16, 16, 17, 26

Offset

1,1

Comments

See A204892 for a discussion and guide to related sequences.

Links

Table of n, a(n) for n=1..73.

Mathematica

s[n_] := s[n] = (1/2) Floor[(n + 1)^2/2];
z1 = 1000; z2 = 80;
Table[s[n], {n, 1, 30}]   (* A002620, quarter-squares *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]   (* A205400 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]   (* A205401 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]        (* A205402 *)
Table[j[n], {n, 1, z2}]        (* A205403 *)
Table[s[k[n]], {n, 1, z2}]     (* A205404 *)
Table[s[j[n]], {n, 1, z2}]     (* A205405 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205406 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A198293 *)

Crossrefs

Cf. A002620, A205400, A204892.

Sequence in context: A164512 A127434 A266475 * A322007 A226107 A365659

Adjacent sequences: A205399 A205400 A205401 * A205403 A205404 A205405

Keyword

nonn

Author

Clark Kimberling, Jan 27 2012

Extensions

Name edited by Clark Kimberling, Dec 06 2021

Status

approved