A190751 - OEIS

%I #8 Apr 09 2021 22:16:25

%S 2,4,8,10,13,18,21,24,27,29,34,38,42,44,47,51,54,58,61,63,66,70,74,78,

%T 80,84,87,91,94,97,100,104,107,111,114,118,120,125,127,129,134,136,

%U 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

%N 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.

%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(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.

%t x=Pi/2;

%t r = Sinh[x]; s = Cosh[x]; t = Tanh[x]; 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}]  (* A190751 *)

%t Table[b[n], {n, 1, 120}]  (* A190752 *)

%t Table[c[n], {n, 1, 120}]  (* A190753 *)

%t Table[d[n], {n, 1, 120}]  (* A190754 *)

%t Table[e[n], {n, 1, 120}]  (* A190755 *)

%t Table[f[n], {n, 1, 120}]  (* A190756 *)

%Y Cf. A190752, A190753, A190754, A190755, A190756.

%K nonn

%O 1,1

%A _Clark Kimberling_, May 18 2011