A206807 Position of 3^n when {2^j} and {3^k} are jointly ranked; complement of A206805.

2, 5, 7, 10, 12, 15, 18, 20, 23, 25, 28, 31, 33, 36, 38, 41, 43, 46, 49, 51, 54, 56, 59, 62, 64, 67, 69, 72, 74, 77, 80, 82, 85, 87, 90, 93, 95, 98, 100, 103, 105, 108, 111, 113, 116, 118, 121, 124, 126, 129, 131, 134, 137, 139, 142, 144, 147, 149, 152, 155

Offset

1,1

Comments

The joint ranking is for j >= 1 and k >= 1, so that the sets {2^j} and {3^k} are disjoint.

Links

Table of n, a(n) for n=1..60.

Formula

a(n) = n + floor(n*log_2(3)).

A206805(n) = n + floor(n*log_3(2)).

a(n) = n + A056576(n). - Michel Marcus, Dec 12 2023

a(n) = A098294(n) + 2*n - 1. - Ruud H.G. van Tol, Jan 22 2024

Example

The joint ranking begins with 2,3,4,8,9,16,27,32,64,81,128,243,256, so that
A206805 = (1,3,4,6,8,9,11,13,...)
A206807 = (2,5,7,10,12,...)

Mathematica

f[n_] := 2^n; g[n_] := 3^n; z = 200;
c = Table[f[n], {n, 1, z}]; s = Table[g[n], {n, 1, z}];
j = Sort[Union[c, s]];
p[n_] := Position[j, f[n]]; q[n_] := Position[j, g[n]];
Flatten[Table[p[n], {n, 1, z}]]           (* A206805 *)
Table[n + Floor[n*Log[3, 2]], {n, 1, 50}] (* A206805 *)
Flatten[Table[q[n], {n, 1, z}]]           (* this sequence *)
Table[n + Floor[n*Log[2, 3]], {n, 1, 50}] (* this sequence as a table *)

Prog

(PARI) a(n) = logint(3^n, 2) + n; \\ Ruud H.G. van Tol, Dec 10 2023

Crossrefs

Cf. A006899, A056576, A098294, A206805, A122437.

Sequence in context: A182771 A109260 A292664 * A182773 A346155 A298869

Adjacent sequences: A206804 A206805 A206806 * A206808 A206809 A206810

Keyword

nonn

Author

Clark Kimberling, Feb 16 2012

Status

approved