A322532 Position of 1/2^n in the sequence of all numbers 1/2^m, 1/3^m, 2/3^m arranged in decreasing order.

2, 4, 6, 9, 11, 13, 16, 18, 20, 22, 24, 27, 29, 31, 34, 36, 38, 40, 42, 45, 47, 49, 52, 54, 56, 59, 61, 63, 65, 67, 70, 72, 74, 77, 79, 81, 83, 85, 88, 90, 92, 95, 97, 99, 102, 104, 106, 108, 110, 113, 115, 117, 120, 122, 124, 126, 128, 131, 133, 135, 138

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. A322533, A322534.

Sequence in context: A329826 A330908 A327213 * A380420 A383009 A284365

Adjacent sequences: A322529 A322530 A322531 * A322533 A322534 A322535

Keyword

nonn

Author

Clark Kimberling, Dec 14 2018

Status

approved