A190739 a(n) = n + [n*s/r] + [n*t/r] + [n*u/r] + [n*v/r] + [n*w/r], where r=sin(x), s=cos(x), t=tan(x), u=csc(x), v=sec(x), w=cot(x), x=Pi/8.

18, 38, 59, 80, 101, 119, 140, 161, 181, 203, 223, 241, 264, 284, 305, 325, 347, 365, 385, 407, 427, 448, 469, 487, 509, 530, 551, 571, 592, 611, 631, 652, 673, 694, 712, 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

Offset

1,1

Comments

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

a(n) = n + [n*s/r] + [n*t/r] + [n*u/r] + [n*v/r] + [n*w/r],

b(n) = [n*r/s] + [n*t/s] + [n*u/s] + [n*v/s] + [n*w/s],

c(n) = [n*r/t] + [n*s/t] + [n*u/t] + [n*v/t] + [n*w/t],

d(n) = n + [n*r/u] + [n*s/u] + [n*t/u] + [n*v/u] + [n*w/u],

e(n) = n + [n*r/v] + [n*s/v] + [n*t/v] + [n*u/v] + [n*w/v],

f(n) = n + [n*r/w] + [n*s/w] + [n*t/w] + [n*u/w] + [n*v/w], where []=floor.

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.

Links

Table of n, a(n) for n=1..63.

Mathematica

x = Pi/8;
r = Sin[x]; s = Cos[x]; t = Tan[x]; u = 1/r; v = 1/s; w = 1/t;
p[n_, h_, k_] := Floor[n*h/k]
a[n_] := n + p[n, s, r] + p[n, t, r] + p[n, u, r] + p[n, v, r] + p[n, w, r]
b[n_] := n + p[n, r, s] + p[n, t, s] + p[n, u, s] + p[n, v, s] + p[n, w, s]
c[n_] := n + p[n, r, t] + p[n, s, t] + p[n, u, t] + p[n, v, t] + p[n, w, t]
d[n_] := n + p[n, r, u] + p[n, s, u] + p[n, t, u] + p[n, v, u] + p[n, w, u]
e[n_] := n + p[n, r, v] + p[n, s, v] + p[n, t, v] + p[n, u, v] + p[n, w, v]
f[n_] := n + p[n, r, w] + p[n, s, w] + p[n, t, w] + p[n, u, w] + p[n, v, w]
Table[a[n], {n, 1, 120}]  (* A190739 *)
Table[b[n], {n, 1, 120}]  (* A190740 *)
Table[c[n], {n, 1, 120}]  (* A190741 *)
Table[d[n], {n, 1, 120}]  (* A190742 *)
Table[e[n], {n, 1, 120}]  (* A190743 *)
Table[f[n], {n, 1, 120}]  (* A190744 *)

Crossrefs

Cf. A190513, A190520.

Sequence in context: A244412 A114698 A119127 * A084585 A132761 A079862

Adjacent sequences: A190736 A190737 A190738 * A190740 A190741 A190742

Keyword

nonn,changed

Author

Clark Kimberling, May 18 2011

Status

approved