A189383 a(n) = n + [n*s/r] + [n*t/r]; r=1, s=1/sqrt(2), t=1/sqrt(3).

1, 4, 6, 8, 10, 13, 15, 17, 20, 22, 24, 26, 29, 31, 33, 36, 38, 40, 42, 45, 47, 49, 52, 53, 56, 59, 61, 63, 65, 68, 69, 72, 75, 77, 79, 81, 84, 85, 88, 91, 92, 95, 97, 100, 101, 104, 107, 108, 111, 113, 116, 118, 120, 123, 124, 127, 129, 132, 134, 136, 139, 140, 143, 145, 147, 150, 152, 155, 156, 159, 161, 163, 166, 168, 171, 172, 175, 178, 179, 182, 184, 186, 188, 191

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 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 + [n*s/r] + [n*t/r],

b(n) = n + [n*r/s] + [n*t/s],

c(n) = n + [n*r/t] + [n*s/t], where []=floor.

Taking r=1, s=1/sqrt(2), t=1/sqrt(3) gives

a=A189383, b=A189384, c=A189385.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Mathematica

r=1; s=2^(-1/2); t=3^(-1/2);
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}]  (*A189383*)
Table[b[n], {n, 1, 120}]  (*A189384*)
Table[c[n], {n, 1, 120}]  (*A189385*)

Crossrefs

Cf. A189384, A189385, A188386, A189361, A189386, A189395.

Sequence in context: A190330 A186291 A189473 * A287106 A087684 A182091

Adjacent sequences: A189380 A189381 A189382 * A189384 A189385 A189386

Keyword

nonn

Author

Clark Kimberling, Apr 21 2011

Status

approved