A330180 a(n) = n + floor(nr/t) + floor(ns/t), where r = e - 1, s = e, t = e + 1.

1, 3, 6, 7, 10, 12, 15, 16, 19, 21, 24, 25, 28, 30, 31, 34, 36, 39, 40, 43, 45, 48, 49, 52, 54, 57, 58, 60, 63, 64, 67, 69, 72, 73, 76, 78, 81, 82, 85, 87, 88, 91, 93, 96, 97, 100, 102, 105, 106, 109, 111, 114, 115, 117, 120, 121, 124, 126, 129, 130, 133

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

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

Links

Table of n, a(n) for n=1..61.

Formula

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

Mathematica

r = E - 1; s = E; t = E + 1;
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}]  (* A330179 *)
Table[b[n], {n, 1, 120}]  (* A016789 *)
Table[c[n], {n, 1, 120}]  (* A330180 *)

Crossrefs

Cf. A016789, A330179.

Sequence in context: A096604 A184873 A184903 * A330184 A008912 A101885

Adjacent sequences: A330177 A330178 A330179 * A330181 A330182 A330183

Keyword

nonn,easy

Author

Clark Kimberling, Jan 05 2020

Status

approved