A189361 a(n) = n + floor(n*s/r) + floor(n*t/r); r=1, s=sqrt(2), t=sqrt(3).

3, 7, 12, 15, 20, 24, 28, 32, 36, 41, 45, 48, 53, 57, 61, 65, 70, 74, 77, 82, 86, 91, 94, 98, 103, 107, 111, 115, 120, 123, 127, 132, 136, 140, 144, 148, 153, 156, 161, 165, 169, 173, 177, 182, 185, 190, 194, 198, 202, 206, 211, 215, 218, 223, 227, 231, 235, 240, 244, 247, 252, 256, 261, 264, 268, 273, 277, 281, 285, 289, 293, 297, 302, 306, 310, 314, 318, 323, 326, 331, 335, 339, 343

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=sqrt(2), t=sqrt(3) gives

a=A189361, b=A189362, c=A189363.

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}]  (*A189361*)
Table[b[n], {n, 1, 120}]  (*A189362*)
Table[c[n], {n, 1, 120}]  (*A189363*)

Crossrefs

Cf. A189362, A189363, A184812, A189383, A189386, A189395.

Sequence in context: A310232 A310233 A189458 * A107790 A310234 A310235

Adjacent sequences: A189358 A189359 A189360 * A189362 A189363 A189364

Keyword

nonn

Author

Clark Kimberling, Apr 20 2011

Status

approved