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A190745
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n+[ns/r]+[nt/r]+[nu/r]+[nv/r]+[nw/r], where r=sinh(1), s=cosh(1), t=tanh(1), u=csch(1), v=sech(1), w=coth(1).
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6
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3, 9, 13, 19, 24, 29, 35, 40, 45, 52, 57, 61, 68, 73, 77, 83, 89, 94, 99, 105, 110, 115, 120, 126, 131, 137, 142, 148, 152, 158, 164, 169, 174, 179, 185, 191, 195, 200, 207, 211, 216, 223, 228, 233, 239, 244, 249, 255, 260, 265, 270, 276, 282, 286, 292, 297, 302, 307, 313, 319, 325, 330, 334, 341, 346, 350, 355, 362, 367, 372, 379
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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(1), s=cosh(1), t=tanh(1), u=csch(1), v=sech(1), w=coth(1) gives a=A190745, b=A190746, c=A190747, d=A190748, e=A190749, f=A190750.
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LINKS
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MATHEMATICA
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r = Sinh[1]; s = Cosh[1]; t = Tanh[1]; 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}] (*A190745*)
Table[b[n], {n, 1, 120}] (*A190746*)
Table[c[n], {n, 1, 120}] (*A190747*)
Table[d[n], {n, 1, 120}] (*A190748*)
Table[e[n], {n, 1, 120}] (*A190749*)
Table[f[n], {n, 1, 120}] (*A190750*)
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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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