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A190504 n+[ns/r]+[nt/r]+[nu/r]; r=golden ratio, s=r+1, t=r+2, u=r+3. 4
6, 14, 21, 29, 38, 45, 52, 59, 68, 76, 83, 91, 100, 106, 114, 121, 130, 138, 145, 153, 159, 168, 176, 183, 191, 200, 207, 214, 221, 230, 238, 245, 253, 262, 268, 276, 283, 291, 300, 307, 315, 321, 330, 338, 345, 353, 362, 368, 376, 383, 392, 400, 407, 415, 421, 430, 438, 445, 453, 462, 469, 476, 483, 492, 500, 507, 515, 524, 530 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This is one of four sequences that partition the positive integers.  In general, suppose that r, s, t, u are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1, {h/u: h>=1} are pairwise disjoint.  Let a(n) be the rank of n/r when all the numbers in the four sets are jointly ranked.  Define b(n), c(n), d(n) as the ranks of n/s, n/t, n/u, respectively.  It is easy to prove that

a(n)=n+[ns/r]+[nt/r]+[nu/r],

b(n)=n+[nr/s]+[nt/s]+[nu/s],

c(n)=n+[nr/t]+[ns/t]+[nu/t],

d(n)=n+[nr/u]+[ns/u]+[nt/u], where []=floor.

Taking r=golden ratio, s=r+1, t=r+2, u=r+3 gives

a=A190504, b=A190505, c=A190506, d=A190507.

LINKS

Table of n, a(n) for n=1..69.

MATHEMATICA

r=GoldenRatio; s=r+1; t=r+2; u=r+3;

a[n_] := n + Floor[n*s/r] + Floor[n*t/r]+Floor[n*u/r];

b[n_] := n + Floor[n*r/s] + Floor[n*t/s]+Floor[n*u/s];

c[n_] := n + Floor[n*r/t] + Floor[n*s/t]+Floor[n*u/t];

d[n_] := n + Floor[n*r/u] + Floor[n*s/u]+Floor[n*t/u];

Table[a[n], {n, 1, 120}]  (*A190504*)

Table[b[n], {n, 1, 120}]  (*A190505*)

Table[c[n], {n, 1, 120}]  (*A190506*)

Table[d[n], {n, 1, 120}]  (*A190507*)

CROSSREFS

Cf. A190505, A190506, A190507 (the other three sequences in the partition of N).

Sequence in context: A063299 A184924 A110223 * A175582 A182081 A125086

Adjacent sequences:  A190501 A190502 A190503 * A190505 A190506 A190507

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 11 2011

STATUS

approved

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Last modified July 8 08:08 EDT 2020. Contains 335520 sequences. (Running on oeis4.)