A330214 - OEIS

%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