%I #4 Jan 05 2020 12:59:43
%S 1,3,4,7,9,10,12,15,16,18,19,22,24,25,27,30,31,33,36,37,39,40,43,45,
%T 46,48,51,52,54,55,58,60,61,63,66,67,69,72,73,75,76,79,81,82,84,87,88,
%U 90,91,94,96,97,100,102,103,105,108,109,111,112,115,117,118
%N a(n) = n + floor(nr/t) + floor(ns/t), where r = e - 2, s = e - 1, t = 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 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 = e - 2, s = e - 1, t = e yields
%C a=A330177, b=A016789, c=A330178.
%F a(n) = n + floor(nr/t) + floor(ns/t), where r = e - 2, s = e - 1, t = e.
%t r = E - 2; s = E - 1; t = 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}] (* A330177 *)
%t Table[b[n], {n, 1, 120}] (* A016789 *)
%t Table[c[n], {n, 1, 120}] (* A330178 *)
%Y Cf. A016789, A330177.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jan 05 2020