OFFSET
1,1
COMMENTS
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],
MATHEMATICA
x = 2Pi/5;
r = Sin[x]; s = Cos[x]; t = Tan[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}] (* A190519 *)
Table[b[n], {n, 1, 120}] (* A190520 *)
Table[c[n], {n, 1, 120}] (* A190521 *)
Table[d[n], {n, 1, 120}] (* A190522 *)
Table[e[n], {n, 1, 120}] (* A190523 *)
Table[f[n], {n, 1, 120}] (* A190524 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 11 2011
STATUS
approved