A189395 a(n) = n + [n*s/r] + [n*t/r]; r=1, s=1/sqrt(2), t=sqrt(3).

2, 6, 10, 12, 16, 20, 23, 26, 30, 34, 37, 40, 44, 47, 50, 54, 58, 61, 64, 68, 71, 75, 78, 81, 85, 89, 92, 95, 99, 102, 105, 109, 113, 116, 119, 123, 127, 129, 133, 137, 140, 143, 147, 151, 153, 157, 161, 164, 167, 171, 175, 178, 181, 185, 188, 191, 195, 199, 202, 205, 209, 212, 216, 219, 222, 226, 230, 233, 236, 240, 243, 246, 250, 254, 257, 260, 264, 268, 270, 274, 278, 281, 284

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 + [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=sqrt(3) gives

a=A189395, b=A189396, c=A189397.

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}]  (*A189395*)
Table[b[n], {n, 1, 120}]  (*A189396*)
Table[c[n], {n, 1, 120}]  (*A189397*)

Crossrefs

Cf. A189396, A189397, A189361, A189383, A189386.

Sequence in context: A055743 A354424 A189680 * A190003 A357702 A263309

Adjacent sequences: A189392 A189393 A189394 * A189396 A189397 A189398

Keyword

nonn

Author

Clark Kimberling, Apr 21 2011

Status

approved