A330178 a(n) = n + floor(nr/t) + floor(ns/t), where r = e - 2, s = e - 1, t = e.

1, 3, 4, 7, 9, 10, 12, 15, 16, 18, 19, 22, 24, 25, 27, 30, 31, 33, 36, 37, 39, 40, 43, 45, 46, 48, 51, 52, 54, 55, 58, 60, 61, 63, 66, 67, 69, 72, 73, 75, 76, 79, 81, 82, 84, 87, 88, 90, 91, 94, 96, 97, 100, 102, 103, 105, 108, 109, 111, 112, 115, 117, 118

Offset

1,2

Comments

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

a(n)=n+[ns/r]+[nt/r],

b(n)=n+[nr/s]+[nt/s],

c(n)=n+[nr/t]+[ns/t], where []=floor.

Taking r = e - 2, s = e - 1, t = e yields

a=A330177, b=A016789, c=A330178.

Links

Table of n, a(n) for n=1..63.

Formula

a(n) = n + floor(nr/t) + floor(ns/t), where r = e - 2, s = e - 1, t = e.

Mathematica

r = E - 2; s = E - 1; t = E;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[a[n], {n, 1, 120}]  (* A330177 *)
Table[b[n], {n, 1, 120}]  (* A016789 *)
Table[c[n], {n, 1, 120}]  (* A330178 *)

Crossrefs

Cf. A016789, A330177.

Sequence in context: A233010 A263488 A344416 * A059010 A066928 A032726

Adjacent sequences: A330175 A330176 A330177 * A330179 A330180 A330181

Keyword

nonn,easy

Author

Clark Kimberling, Jan 05 2020

Status

approved