A189792 n + [nr/s] + [nt/s] with r = 1, s = (sin(1))^2, t = (cos(1))^2.

2, 4, 8, 10, 14, 16, 18, 22, 24, 28, 30, 32, 36, 38, 42, 44, 48, 50, 52, 56, 58, 62, 64, 66, 70, 72, 76, 78, 80, 84, 86, 90, 92, 96, 98, 100, 104, 106, 110, 112, 114, 118, 120, 124, 126, 128, 132, 134, 138, 140, 144, 146, 148, 152, 154, 158, 160, 162, 166, 168, 172, 174, 176, 180, 182, 186, 188, 192, 194, 196, 200, 202, 206, 208, 210, 214, 216, 220, 222, 224, 228, 230, 234, 236

Offset

1,1

Comments

This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that

a(n) = n + [ns/r] + [nt/r],

b(n) = n + [nr/s] + [nt/s],

c(n) = n + [nr/t] + [ns/t], where []=floor.

Taking r = 1, s = (sin(1))^2, t = (cos(1))^2 gives a = A005408, b = A189792, c = A189793.

Differs from A286909 first in a(29) = 80, while A286909(29) = 82. - M. F. Hasler, Apr 13 2018

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Formula

b(n)=n+[n*u^2]+[n*v^2], where u=csc(1), v=cot(1).

Mathematica

r=1; s=Sin[1]^2; t=Cos[1]^2;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 120}]   (* A005408 *)
Table[b[n], {n, 1, 120}]   (* A189792 *)
Table[c[n], {n, 1, 120}]   (* A189793 *)
Table[b[n]/2, {n, 1, 120}] (* A189794 *)
Table[c[n]/2, {n, 1, 120}] (* A189795 *)
Table[n+Floor[n/Sin[1]^2]+Floor[(n Cos[1]^2)/Sin[1]^2], {n, 90}] (* Harvey P. Dale, Mar 08 2023 *)

Crossrefs

Cf. A005408, A189793, A189794, A189795.

Sequence in context: A350804 A285626 A284853 * A286909 A356139 A283967

Adjacent sequences: A189789 A189790 A189791 * A189793 A189794 A189795

Keyword

nonn

Author

Clark Kimberling, Apr 27 2011

Status

approved