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A330180
a(n) = n + floor(nr/t) + floor(ns/t), where r = e - 1, s = e, t = e + 1.
2
1, 3, 6, 7, 10, 12, 15, 16, 19, 21, 24, 25, 28, 30, 31, 34, 36, 39, 40, 43, 45, 48, 49, 52, 54, 57, 58, 60, 63, 64, 67, 69, 72, 73, 76, 78, 81, 82, 85, 87, 88, 91, 93, 96, 97, 100, 102, 105, 106, 109, 111, 114, 115, 117, 120, 121, 124, 126, 129, 130, 133
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 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+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where []=floor.
Taking r = e - 1, s = e, t = e + 1 yields
FORMULA
a(n) = n + floor(nr/t) + floor(ns/t), where r = e - 1, s = e, t = e + 1.
MATHEMATICA
r = E - 1; s = E; t = E + 1;
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}] (* A330179 *)
Table[b[n], {n, 1, 120}] (* A016789 *)
Table[c[n], {n, 1, 120}] (* A330180 *)
CROSSREFS
Sequence in context: A096604 A184873 A184903 * A330184 A008912 A101885
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 05 2020
STATUS
approved