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.
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 27 2011
STATUS
approved