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n+[ns/r]+[nt/r]; r=1, s=e, t=e^2.
5

%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