%I
%S 2,5,8,11,14,17,20,23,26,30,32,35,38,41,44,47,50,53,56,60,63,65,68,71,
%T 74,77,80,83,87,90,93,95,98,101,104,107,110,113,117,120,123,126,128,
%U 131,134,137,140,143,147,150,153,156,159,161,164,167,170,174,177,180,183,186,189,191,194,197,200,204,207,210,213,216,219,222,224,227,230,234,237,240,243,246,249
%N n+[ns/r]+[nt/r]; r=1, s=e/3, t=3/e.
%C 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
%C a(n)=n+[ns/r]+[nt/r],
%C b(n)=n+[nr/s]+[nt/s],
%C c(n)=n+[nr/t]+[ns/t], where []=floor.
%C Taking r=1, s=e/3, t=3/e gives
%C a=A189512, b=A189513, c=A189514.
%t r=1; s=E/3; t=3/E;
%t a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
%t b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
%t c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
%t Table[a[n], {n, 1, 120}] (*A189512*)
%t Table[b[n], {n, 1, 120}] (*A189513*)
%t Table[c[n], {n, 1, 120}] (*A189514*)
%Y Cf. A189513, A189514.
%K nonn
%O 1,1
%A _Clark Kimberling_, Apr 23 2011
