A322533 Position of 1/3^n in the sequence of all numbers 1/2^m, 1/3^m, 2/3^m arranged in decreasing order.

3, 7, 10, 14, 17, 21, 25, 28, 32, 35, 39, 43, 46, 50, 53, 57, 60, 64, 68, 71, 75, 78, 82, 86, 89, 93, 96, 100, 103, 107, 111, 114, 118, 121, 125, 129, 132, 136, 139, 143, 146, 150, 154, 157, 161, 164, 168, 172, 175, 179, 182, 186, 190, 193, 197, 200, 204

Offset

1,1

Comments

Every positive integer is in exactly one of the sequences A322532, A322533, A322534.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Formula

Position of 1/2^n: n + floor(n log(2)/log(3)) + floor((n + 1) log(2)/log(3))

Position of 1/3^n: 2n - 2 + floor(n log(3)/log(2))

Position of 2/3^n: 2n + floor(n log(3)/log(2))

Example

In the decreasing sequence 2/3, 1/2, 1/3, 1/4, 2/9, 1/8, 1/9, 2/27, 1/16, ..., the positions of 1/2, 1/4, 1/8, 1/6, are 2,4,6,9; the positions of 1/3, 1/9, 1/27,... are 3,7,10,14,...; the positions of 2/3, 2/9,2/27,... are 1,5,8,12,...

Mathematica

a[n_] := n + Floor[n Log[2]/Log[3]] + Floor[(n + 1) Log[2]/Log[3]];
b[n_] := 2 n - 2 + Floor[n Log[3]/Log[2]];
c[n_] := 2 n + Floor[n Log[3]/Log[2]];
Table[a[n], {n, 1, 120}]  (* A322532 *)
Table[b[n], {n, 1, 120}]  (* A322533 *)
Table[c[n], {n, 1, 120}]  (* A322534 *)

Crossrefs

Cf. A322532, A322534.

Sequence in context: A047355 A248522 A098005 * A189999 A310188 A171983

Adjacent sequences: A322530 A322531 A322532 * A322534 A322535 A322536

Keyword

nonn

Author

Clark Kimberling, Dec 14 2018

Status

approved