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A190513
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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/5.
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8
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9, 19, 31, 41, 52, 64, 74, 85, 97, 107, 118, 129, 140, 151, 162, 173, 184, 195, 205, 216, 227, 239, 249, 261, 271, 282, 294, 304, 313, 327, 337, 348, 359, 370, 381, 392, 402, 412, 425, 436, 447, 457, 469, 480, 490, 501, 513, 523, 534, 545, 557, 567, 578, 589, 599, 612, 621, 631, 644, 655, 665, 677, 687, 699
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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/5, gives a=A190513, b=A190514, c=A190515, d=A190516, e=A190517, f=A190518.
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LINKS
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MATHEMATICA
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x = Pi/5;
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}] (*A190513*)
Table[b[n], {n, 1, 120}] (*A190514*)
Table[c[n], {n, 1, 120}] (*A190515*)
Table[d[n], {n, 1, 120}] (*A190516*)
Table[e[n], {n, 1, 120}] (*A190517*)
Table[f[n], {n, 1, 120}] (*A190518*)
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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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