A330181 a(n) = n + floor(ns/r) + floor(nt/r), where r = Pi - 1, s = Pi, t = Pi + 1.

3, 7, 12, 16, 21, 25, 30, 34, 39, 43, 48, 52, 57, 61, 66, 69, 73, 78, 82, 87, 91, 96, 100, 105, 109, 114, 118, 123, 127, 132, 135, 139, 144, 148, 153, 157, 162, 166, 171, 175, 180, 184, 189, 193, 198, 201, 205, 210, 214, 219, 223, 228, 232, 237, 241, 246

Offset

1,1

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 = Pi - 1, s = Pi, t = Pi + 1 yields a=A330181, b=A016789, c=A330182.

Links

Table of n, a(n) for n=1..56.

Formula

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

Mathematica

r = Pi - 1; s = Pi; t = Pi + 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}]  (* A330181 *)
Table[b[n], {n, 1, 120}]  (* A016789 *)
Table[c[n], {n, 1, 120}]  (* A330182 *)
Table[n+Floor[n Pi/(Pi-1)]+Floor[n (Pi+1)/(Pi-1)], {n, 60}] (* Harvey P. Dale, Jan 28 2023 *)

Crossrefs

Cf. A016789, A330182.

Sequence in context: A310243 A286923 A330185 * A184917 A084582 A072098

Adjacent sequences: A330178 A330179 A330180 * A330182 A330183 A330184

Keyword

nonn,easy

Author

Clark Kimberling, Jan 05 2020

Status

approved