A322534 Position of 2/3^n in the sequence of all numbers 1/2^m, 1/3^m, 2/3^m arranged in decreasing order.

1, 5, 8, 12, 15, 19, 23, 26, 30, 33, 37, 41, 44, 48, 51, 55, 58, 62, 66, 69, 73, 76, 80, 84, 87, 91, 94, 98, 101, 105, 109, 112, 116, 119, 123, 127, 130, 134, 137, 141, 144, 148, 152, 155, 159, 162, 166, 170, 173, 177, 180, 184, 188, 191, 195, 198, 202, 205

Offset

1,2

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: A214858 A186276 A047383 * A314402 A133795 A247060

Adjacent sequences: A322531 A322532 A322533 * A322535 A322536 A322537

Keyword

nonn

Author

Clark Kimberling, Dec 14 2018

Status

approved