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A190739
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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.
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6
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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
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OFFSET
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1,1
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COMMENTS
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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 + [ns/r] + [nt/r] + [nu/r] + [nv/r] + [nw/r],
b(n) = [nr/s] + [nt/s] + [nu/s] + [nv/s] + [nw/s],
c(n) = [nr/t] + [ns/t] + [nu/t] + [nv/t] + [nw/t],
d(n) = n + [nr/u] + [ns/u] + [nt/u] + [nv/u] + [nw/u],
e(n) = n + [nr/v] + [ns/v] + [nt/v] + [nu/v] + [nw/v],
f(n) = n + [nr/w] + [ns/w] + [nt/w] + [nu/w] + [nv/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.
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LINKS
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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