A189370 a(n) = n + [n*s/r] + [n*t/r]; r=1, s=sqrt(2), t=sqrt(5).

4, 8, 13, 17, 23, 27, 31, 36, 41, 46, 50, 54, 60, 64, 69, 73, 79, 83, 87, 92, 96, 102, 106, 110, 115, 120, 125, 129, 134, 139, 143, 148, 152, 158, 162, 166, 171, 175, 181, 185, 189, 194, 199, 204, 208, 213, 218, 222, 227, 231, 237, 241, 245, 250, 254, 260, 264, 269, 273, 278, 283, 287, 292, 297, 301, 306, 310, 316, 320, 324, 329, 333, 339, 343, 348, 352, 357, 362, 366, 371, 376, 380, 385

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

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

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

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

Taking r=1, s=sqrt(2), t=sqrt(5) gives f=A189370, g=A189371, h=A189372.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Mathematica

r = 1; s = Sqrt[2]; t = Sqrt[5];
f[n_] := n + Floor[n*s/r] + Floor[n*t/r];
g[n_] := n + Floor[n*r/s] + Floor[n*t/s];
h[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[f[n], {n, 1, 120}]  (* A189370 *)
Table[g[n], {n, 1, 120}]  (* A189371 *)
Table[h[n], {n, 1, 120}]  (* A189372 *)

Prog

(PARI) for(n=1, 100, print1(n + floor(n*sqrt(2)) + floor(n*sqrt(5)), ", ")) \\ G. C. Greubel, Apr 20 2018
(Magma) [n + Floor(n*Sqrt(2)) + Floor(n*Sqrt(5)): n in [1..100]]; // G. C. Greubel, Apr 20 2018

Crossrefs

Cf. A189371, A189372.

Sequence in context: A311989 A311990 A311991 * A311992 A311993 A311994

Adjacent sequences: A189367 A189368 A189369 * A189371 A189372 A189373

Keyword

nonn

Author

Clark Kimberling, Apr 20 2011

Status

approved