A190519 - OEIS

%I #10 Feb 15 2025 03:06:47

%S 8,16,26,35,45,54,63,72,82,93,101,110,121,129,139,148,157,167,177,186,

%T 195,205,214,223,233,242,251,261,270,281,290,298,308,318,327,336,345,

%U 355,365,374,384,392,402,412,421,429,440,450,458,469,478,486,497,505,514,525,534,543,553,563,571,581,590,599,610,618,627,639

%N a(n) = n + [n*s/r] + [n*t/r] + [n*u/r] + [n*v/r] + [n*w/r], where r=sin(x), s=cos(x), t=tan(x), u=csc(x), v=sec(x), w=cot(x), x=2*Pi/5.

%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 + [n*s/r] + [n*t/r] + [n*u/r] + [n*v/r] + [n*w/r],

%C   b(n) = [n*r/s] + [n*t/s] + [n*u/s] + [n*v/s] + [n*w/s],

%C   c(n) = [n*r/t] + [n*s/t] + [n*u/t] + [n*v/t] + [n*w/t],

%C   d(n) = n + [n*r/u] + [n*s/u] + [n*t/u] + [n*v/u] + [n*w/u],

%C   e(n) = n + [n*r/v] + [n*s/v] + [n*t/v] + [n*u/v] + [n*w/v],

%C   f(n) = n + [n*r/w] + [n*s/w] + [n*t/w] + [n*u/w] + [n*v/w], where []=floor. Choosing r=sin(x), s=cos(x), t=tan(x), u=csc(x), v=sec(x), w=cot(x), x=2*Pi/5, gives a=A190519, b=A190520, c=A190521, d=A190522, e=A190523, f=A190524.

%t x = 2Pi/5;

%t r = Sin[x]; s = Cos[x]; t = Tan[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}]  (* A190519 *)

%t Table[b[n], {n, 1, 120}]  (* A190520 *)

%t Table[c[n], {n, 1, 120}]  (* A190521 *)

%t Table[d[n], {n, 1, 120}]  (* A190522 *)

%t Table[e[n], {n, 1, 120}]  (* A190523 *)

%t Table[f[n], {n, 1, 120}]  (* A190524 *)

%Y Cf. A190520, A190521, A190522, A190523, A190524 (the other 5 members of the partition) and A190513-A190518 (based on Pi/5).

%K nonn,changed

%O 1,1

%A _Clark Kimberling_, May 11 2011