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A189785 a(n) = n+floor(n*r/s)+floor(nt/s); r=Pi/2, s=arcsin(5/13), t=arcsin(12/13). 3
6, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 182, 190, 198, 206, 214, 222, 230, 238, 246, 254, 262, 270, 278, 286, 294, 302, 310, 318, 326, 334, 342, 350, 358, 366, 374, 380, 388, 396, 404, 412, 420, 428, 436, 444, 452, 460, 468, 476, 484, 492, 500, 508, 516, 524, 532, 540, 548 (list; graph; refs; listen; history; text; internal format)
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=A005408, b=A189785, c=A189786.  Note that r=s+t.

a(n) first differs from A017137(n-1) at n=48 (a(48)=380 but A017137(47)=382). - Nathaniel Johnston, May 16 2011

LINKS

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

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

Cf. A189786, A189787, A004773.

Sequence in context: A062316 A079299 A043445 * A017137 A211998 A190522

Adjacent sequences:  A189782 A189783 A189784 * A189786 A189787 A189788

KEYWORD

nonn

AUTHOR

Clark Kimberling, Apr 27 2011

STATUS

approved

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Last modified October 20 20:24 EDT 2019. Contains 328273 sequences. (Running on oeis4.)