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

Author

Clark Kimberling, Feb 16 2012

Keywords

• nonn

Comments

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

Examples

			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,...)
		

Crossrefs

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

Programs

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 *)

PARI

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

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