A189753 n+[ns/r]+[nt/r]; r=1, s=arctan(1/3), t=arctan(3).

2, 4, 6, 9, 12, 14, 17, 19, 22, 25, 27, 29, 33, 35, 37, 40, 43, 45, 48, 50, 53, 56, 58, 60, 64, 66, 68, 71, 74, 76, 78, 81, 84, 86, 89, 91, 94, 97, 99, 101, 105, 107, 109, 112, 115, 117, 120, 122, 125, 128, 130, 132, 136, 138, 140, 143, 146, 148, 150, 153, 156, 158, 161, 163, 166, 169, 171, 173, 177, 179, 181, 184, 187, 189, 192, 194, 197, 200, 202, 204, 208, 210

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 pairwise 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=1, s=arctan(1/3), t=arctan(3) gives

a=A189753, b=A189754, c=A189755.

Links

Table of n, a(n) for n=1..82.

Mathematica

r=1; s=ArcTan[1/3]; t=ArcTan[3];
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}]  (*A189753*)
Table[b[n], {n, 1, 120}]  (*A189754*)
Table[c[n], {n, 1, 120}]  (*A189755*)

Crossrefs

Cf. A189754, A189755.

Sequence in context: A225600 A183422 A025057 * A278450 A030763 A143145

Adjacent sequences: A189750 A189751 A189752 * A189754 A189755 A189756

Keyword

nonn

Author

Clark Kimberling, Apr 26 2011

Status

approved