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+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where []=floor.
Taking r=Pi/2, s=arcsin(5/13), t=arcsin(12/13) gives
a(n) first differs from A017137(n-1) at n=48 (a(48)=380 but A017137(47)=382). - Nathaniel Johnston, May 16 2011
MATHEMATICA
r=Pi/2; s=ArcSin[5/13]; t=ArcSin[12/13];
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}] (*A005408*)
Table[b[n], {n, 1, 120}] (*A189785*)
Table[c[n], {n, 1, 120}] (*A189786*)
Table[b[n]/2, {n, 1, 120}] (*A189787*)
Table[c[n]/2, {n, 1, 120}] (*A004773*)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 27 2011
STATUS
approved