A189533 - OEIS

%I #8 Jun 10 2020 07:38:53

%S 1,4,6,9,12,14,17,20,22,25,28,30,33,35,37,40,42,45,48,50,53,56,58,61,

%T 64,66,69,71,73,76,78,81,84,86,89,92,94,97,100,102,105,107,109,112,

%U 114,117,120,122,125,128,130,133,136,138,141,143,145,148,150,153,156,158,161,164,166,169,172,174,176,179,181,184,186,189,192,194,197,200,202,205,208,210,212,215

%N n+[ns/r]+[nt/r]; r=1, s=arcsin(3/5), t=arcsin(4/5).

%C 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

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

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

%C c(n)=n+[nr/t]+[ns/t], where []=floor.

%C Taking r=1, s=arcsin(3/5), t=arcsin(4/5) gives

%C a=A189533, b=A189534, c=A189535.

%t r=1; s=ArcSin[3/5]; t=ArcSin[4/5];

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

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

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

%t Table[a[n], {n, 1, 120}]  (*A189533*)

%t Table[b[n], {n, 1, 120}]  (*A189534*)

%t Table[c[n], {n, 1, 120}]  (*A189535*)

%Y Cf. A189534, A189535.

%K nonn

%O 1,2

%A _Clark Kimberling_, Apr 23 2011

%E a(84) corrected by _Georg Fischer_, Jun 10 2020