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%I #7 Mar 30 2012 18:57:24
%S 10,21,33,43,54,66,77,88,99,110,121,132,144,155,165,177,188,199,210,
%T 221,233,243,254,266,276,288,299,310,321,332,344,354,365,377,388,399,
%U 410,421,433,443,454,466,476,488,499,510,521,532,544,554,565,577,588,599,610,621,632,643,654,666,676,688,699
%N n+[ns/r]+[nt/r]; r=1, s=e, t=e^2.
%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=sqrt(e), t=e^2 gives
%C a=A189402, b=A189403, c=A189404.
%t r=1; s=E; t=E^2;
%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}] (*A189402*)
%t Table[b[n], {n, 1, 120}] (*A189403*)
%t Table[c[n], {n, 1, 120}] (*A189404*)
%Y Cf. A189403, A189404, A189399.
%K nonn
%O 1,1
%A _Clark Kimberling_, Apr 21 2011
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