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+[n*s/r]+[n*t/r]+[n*u/r],
b(n)=n+[n*r/s]+[n*t/s]+[n*u/s],
c(n)=n+[n*r/t]+[n*s/t]+[n*u/t],
d(n)=n+[n*r/u]+[n*s/u]+[n*t/u], where []=floor.
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
KEYWORD
nonn,changed
AUTHOR
Clark Kimberling, May 11 2011
STATUS
approved