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A189383
a(n) = n + [n*s/r] + [n*t/r]; r=1, s=1/sqrt(2), t=1/sqrt(3).
8
1, 4, 6, 8, 10, 13, 15, 17, 20, 22, 24, 26, 29, 31, 33, 36, 38, 40, 42, 45, 47, 49, 52, 53, 56, 59, 61, 63, 65, 68, 69, 72, 75, 77, 79, 81, 84, 85, 88, 91, 92, 95, 97, 100, 101, 104, 107, 108, 111, 113, 116, 118, 120, 123, 124, 127, 129, 132, 134, 136, 139, 140, 143, 145, 147, 150, 152, 155, 156, 159, 161, 163, 166, 168, 171, 172, 175, 178, 179, 182, 184, 186, 188, 191
OFFSET
1,2
COMMENTS
This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
a(n) = n + [n*s/r] + [n*t/r],
b(n) = n + [n*r/s] + [n*t/s],
c(n) = n + [n*r/t] + [n*s/t], where []=floor.
Taking r=1, s=1/sqrt(2), t=1/sqrt(3) gives
LINKS
MATHEMATICA
r=1; s=2^(-1/2); t=3^(-1/2);
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[a[n], {n, 1, 120}] (*A189383*)
Table[b[n], {n, 1, 120}] (*A189384*)
Table[c[n], {n, 1, 120}] (*A189385*)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 21 2011
STATUS
approved