A189781 - OEIS

A189781

a(n) = n + [nr/s] + [nt/s]; r=Pi/2, s=arcsin(8/17), t=arcsin(15/17).

6, 12, 18, 24, 32, 38, 44, 50, 56, 64, 70, 76, 82, 88, 96, 102, 108, 114, 120, 128, 134, 140, 146, 152, 160, 166, 172, 178, 184, 192, 198, 204, 210, 218, 224, 230, 236, 242, 250, 256, 262, 268, 274, 282, 288, 294, 300, 306, 314, 320, 326, 332, 338, 346, 352, 358, 364, 370, 378, 384, 390, 396, 402, 410, 416, 422, 428, 436, 442, 448, 454, 460, 468

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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(8/17), t=arcsin(15/17) gives a=A005408, b=A189781, c=A189782. Note that r=s+t.

LINKS

Table of n, a(n) for n=1..73.

MATHEMATICA

r=Pi/2; s=ArcSin[8/17]; t=ArcSin[15/17];

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*)