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A190751
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a(n) = n + [ns/r] + [nt/r] + [nu/r] + [nv/r] + [nw/r], where r=sinh(x), s=cosh(x), t=tanh(x), u=csch(x), v=sech(x), w=coth(x), where x=Pi/2.
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6
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2, 4, 8, 10, 13, 18, 21, 24, 27, 29, 34, 38, 42, 44, 47, 51, 54, 58, 61, 63, 66, 70, 74, 78, 80, 84, 87, 91, 94, 97, 100, 104, 107, 111, 114, 118, 120, 125, 127, 129, 134, 136, 141, 143, 147, 150, 154, 158, 161, 163, 167, 170, 175, 177, 180, 184, 187, 191, 194, 197, 200, 203, 206, 211, 213, 217, 220, 224, 227, 231, 234, 237, 240, 243
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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=sinh(x), s=cosh(x), t=tanh(x), u=csch(x), v=sech(x), w=coth(x), x=Pi/2 gives a=A190751, b=A190752, c=A190753, d=A190754, e=A190755, f=A190756.
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LINKS
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MATHEMATICA
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x=Pi/2;
r = Sinh[x]; s = Cosh[x]; t = Tanh[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}] (* A190751 *)
Table[b[n], {n, 1, 120}] (* A190752 *)
Table[c[n], {n, 1, 120}] (* A190753 *)
Table[d[n], {n, 1, 120}] (* A190754 *)
Table[e[n], {n, 1, 120}] (* A190755 *)
Table[f[n], {n, 1, 120}] (* A190756 *)
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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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