%I #4 Jan 05 2020 13:01:06
%S 2,5,8,10,13,16,18,21,24,26,30,33,36,38,41,44,46,49,52,54,58,61,63,66,
%T 69,72,74,77,80,82,86,89,91,94,97,99,102,105,108,110,114,117,119,122,
%U 125,127,130,133,135,138,142,145,147,150,153,155,158,161,163
%N a(n) = n + floor(nr/s) + floor(nt/s), where r = log(2), s = 1, t = log(3).
%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 = log(2), s = 1, t = log(3) yields
%C a=A330213, b=A330214, c=A330215.
%F a(n) = n + floor(nr/s) + floor(nt/s), where r = log(2), s = 1, t = log(3)
%t r = Log[2]; s = tau; t = Log[3];
%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}] (* A330213 *)
%t Table[b[n], {n, 1, 120}] (* A330214 *)
%t Table[c[n], {n, 1, 120}] (* A330215 *)
%Y Cf. A330213, A330215.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Jan 05 2020