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%I #5 Mar 30 2012 18:57:29
%S 3,9,13,19,24,29,35,40,45,52,57,61,68,73,77,83,89,94,99,105,110,115,
%T 120,126,131,137,142,148,152,158,164,169,174,179,185,191,195,200,207,
%U 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
%N 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).
%C 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
%C a(n)=n+[ns/r]+[nt/r]+[nu/r]+[nv/r]+[nw/r],
%C b(n)=[nr/s]+[nt/s]+[nu/s]+[nv/s]+[nw/s],
%C c(n)=[nr/t]+[ns/t]+[nu/t]+[nv/t]+[nw/t],
%C d(n)=n+[nr/u]+[ns/u]+[nt/u]+[nv/u]+[nw/u],
%C e(n)=n+[nr/v]+[ns/v]+[nt/v]+[nu/v]+[nw/v],
%C f(n)=n+[nr/w]+[ns/w]+[nt/w]+[nu/w]+[nv/w], where []=floor.
%C 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.
%t r = Sinh[1]; s = Cosh[1]; t = Tanh[1]; u = 1/r; v = 1/s; w = 1/t;
%t p[n_, h_, k_] := Floor[n*h/k]
%t a[n_] := n + p[n, s, r] + p[n, t, r] + p[n, u, r] + p[n, v, r] + p[n, w, r]
%t b[n_] := n + p[n, r, s] + p[n, t, s] + p[n, u, s] + p[n, v, s] + p[n, w, s]
%t c[n_] := n + p[n, r, t] + p[n, s, t] + p[n, u, t] + p[n, v, t] + p[n, w, t]
%t d[n_] := n + p[n, r, u] + p[n, s, u] + p[n, t, u] + p[n, v, u] + p[n, w, u]
%t e[n_] := n + p[n, r, v] + p[n, s, v] + p[n, t, v] + p[n, u, v] + p[n, w, v]
%t f[n_] := n + p[n, r, w] + p[n, s, w] + p[n, t, w] + p[n, u, w] + p[n, v, w]
%t Table[a[n], {n, 1, 120}] (*A190745*)
%t Table[b[n], {n, 1, 120}] (*A190746*)
%t Table[c[n], {n, 1, 120}] (*A190747*)
%t Table[d[n], {n, 1, 120}] (*A190748*)
%t Table[e[n], {n, 1, 120}] (*A190749*)
%t Table[f[n], {n, 1, 120}] (*A190750*)
%Y Cf. A190746-A190750.
%K nonn
%O 1,1
%A _Clark Kimberling_, May 18 2011
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