A330179 - OEIS

%I #4 Jan 05 2020 12:59:48

%S 4,9,13,18,22,27,33,37,42,46,51,55,61,66,70,75,79,84,90,94,99,103,108,

%T 112,118,123,127,132,136,141,147,151,156,160,165,169,175,180,184,189,

%U 193,198,204,208,213,217,222,226,232,237,241,246,250,255,261,265

%N a(n) = n + floor(ns/r) + floor(nt/r), where r = e - 1, s = e, t = e + 1.

%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 - 1, s = e, t = e + 1 yields

%C a=A330179, b=A016789, c=A330180.

%F a(n) = n + floor(ns/r) + floor(nt/r), where r = e - 1, s = e, t = e + 1.

%t r = E - 1; s = E; t = E + 1;

%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}]  (* A330179 *)

%t Table[b[n], {n, 1, 120}]  (* A016789 *)

%t Table[c[n], {n, 1, 120}]  (* A330180 *)

%Y Cf. A016789, A330180.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jan 05 2020