A189787 -id:A189787 - cp's OEIS frontend

A189785 a(n) = n+floor(n*r/s)+floor(nt/s); r=Pi/2, s=arcsin(5/13), t=arcsin(12/13).

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

Offset: 1

Clark Kimberling, Apr 27 2011

nonn

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

Cf. A189786, A189787, A004773.

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*)

A189786 a(n) = n + [nr/t] + [ns/t]; r=Pi/2, s=arcsin(5/13), t=arcsin(12/13).

Original entry on oeis.org

2, 4, 8, 10, 12, 16, 18, 20, 24, 26, 28, 32, 34, 36, 40, 42, 44, 48, 50, 52, 56, 58, 60, 64, 66, 68, 72, 74, 76, 80, 82, 84, 88, 90, 92, 96, 98, 100, 104, 106, 108, 112, 114, 116, 120, 122, 124, 128, 130, 132, 136, 138, 140, 144, 146, 148, 152, 154, 156, 160, 162, 164, 168, 170, 172, 176, 178, 180, 184, 186, 188, 192, 194, 196, 200, 202, 204

Offset: 1

Clark Kimberling, Apr 27 2011

See A189785.
Conjecture: Sequence consists of all the positive even numbers except numbers of the form 8*x+6, x >= 0. - Harvey P. Dale, Dec 07 2018
Contains numbers like a(143)=382, a(146)=390, a(149)=398, a(152)=406,... which are not in A047464. - R. J. Mathar, Aug 25 2025
For n<143, a(n) = n+A047217(n+1), but then this formula becomes invalid. - R. J. Mathar, Aug 25 2025

Cf. A189785, A189787.

(See A189785.)
With[{t=ArcSin[12/13]},Table[n+Floor[(n*Pi/2)/t]+Floor[(n*ArcSin[5/13])/t],{n,80}]] (* Harvey P. Dale, Dec 07 2018 *)

cp's OEIS Frontend

A189785 a(n) = n+floor(n*r/s)+floor(nt/s); r=Pi/2, s=arcsin(5/13), t=arcsin(12/13).

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