A190739 - OEIS

%I #8 Apr 09 2021 22:15:59

%S 18,38,59,80,101,119,140,161,181,203,223,241,264,284,305,325,347,365,

%T 385,407,427,448,469,487,509,530,551,571,592,611,631,652,673,694,712,

%U 733,755,775,797,817,835,856,877,898,918,940,958,978,1001,1021,1042,1063,1081,1102,1122,1144,1164,1185,1204,1224,1246,1267,1288

%N a(n) = n + [ns/r] + [nt/r] + [nu/r] + [nv/r] + [nw/r], where r=sin(x), s=cos(x), t=tan(x), u=csc(x), v=sec(x), w=cot(x), x=Pi/8.

%C This is one of six sequences that partition the positive integers. In general, suppose that r, s, t, u, v, w are positive real numbers for which the sets {i/r : i>=1}, {j/s : j>=1}, {k/t : k>=1, {h/u : h>=1}, {p/v : p>=1}, {q/w : q>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the six sets are jointly ranked. Define b(n), c(n), d(n), e(n), f(n) as the ranks of n/s, n/t, n/u, n/v, n/w respectively. It is easy to prove that

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

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

%C   c(n) = [nr/t] + [ns/t] + [nu/t] + [nv/t] + [nw/t],

%C   d(n) = n + [nr/u] + [ns/u] + [nt/u] + [nv/u] + [nw/u],

%C   e(n) = n + [nr/v] + [ns/v] + [nt/v] + [nu/v] + [nw/v],

%C   f(n) = n + [nr/w] + [ns/w] + [nt/w] + [nu/w] + [nv/w], where []=floor.

%C Choosing r=sin(x), s=cos(x), t=tan(x), u=csc(x), v=sec(x), w=cot(x), x=Pi/8, gives a=A190739, b=A190740, c=A190741, d=A190742, e=A190743, f=A190744.

%t x = Pi/8;

%t r = Sin[x]; s = Cos[x]; t = Tan[x]; u = 1/r; v = 1/s; w = 1/t;

%t p[n_, h_, k_] := Floor[n*h/k]

%t a[n_] := n + p[n, s, r] + p[n, t, r] + p[n, u, r] + p[n, v, r] + p[n, w, r]

%t b[n_] := n + p[n, r, s] + p[n, t, s] + p[n, u, s] + p[n, v, s] + p[n, w, s]

%t c[n_] := n + p[n, r, t] + p[n, s, t] + p[n, u, t] + p[n, v, t] + p[n, w, t]

%t d[n_] := n + p[n, r, u] + p[n, s, u] + p[n, t, u] + p[n, v, u] + p[n, w, u]

%t e[n_] := n + p[n, r, v] + p[n, s, v] + p[n, t, v] + p[n, u, v] + p[n, w, v]

%t f[n_] := n + p[n, r, w] + p[n, s, w] + p[n, t, w] + p[n, u, w] + p[n, v, w]

%t Table[a[n], {n, 1, 120}]  (* A190739 *)

%t Table[b[n], {n, 1, 120}]  (* A190740 *)

%t Table[c[n], {n, 1, 120}]  (* A190741 *)

%t Table[d[n], {n, 1, 120}]  (* A190742 *)

%t Table[e[n], {n, 1, 120}]  (* A190743 *)

%t Table[f[n], {n, 1, 120}]  (* A190744 *)

%Y Cf. A190513, A190520.

%K nonn

%O 1,1

%A _Clark Kimberling_, May 18 2011