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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
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
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 11 2011
STATUS
approved