A206812 Position of 2^n in joint ranking of {2^i}, {3^j}, {5^k}.

1, 3, 5, 7, 10, 11, 14, 16, 17, 20, 21, 24, 26, 28, 30, 32, 34, 36, 38, 40, 43, 44, 46, 49, 50, 53, 55, 57, 59, 60, 63, 65, 67, 69, 72, 73, 75, 77, 79, 82, 83, 86, 88, 89, 92, 94, 96, 98, 100, 102, 104, 106, 108, 111, 112, 115, 116, 118, 121, 122, 125, 127, 129

Offset: 1

Author

Clark Kimberling, Feb 17 2012

Keywords

• nonn

Comments

The exponents i,j,k range through the set N of positive integers, so that the position sequences (this sequence for 2^n, A206813 for 3^n, A206814 for 5^n) partition N.

Examples

			The joint ranking begins with 2,3,4,5,8,9,16,25,27,32,64,81,125,128,243 so that
this sequence = (1,3,5,7,10,11,...)
A206813 = (2,6,9,12,15,...)
A206814 = (4,8,13,18,23,...)
		

Crossrefs

Cf. A206805, A206813, A206814.

Programs

Mathematica

f[1, n_] := 2^n; f[2, n_] := 3^n;
f[3, n_] := 5^n; z = 1000;
d[n_, b_, c_] := Floor[n*Log[b, c]];
t[k_] := Table[f[k, n], {n, 1, z}];
t = Sort[Union[t[1], t[2], t[3]]];
p[k_, n_] := Position[t, f[k, n]];
Flatten[Table[p[1, n], {n, 1, z/8}]] (* A206812 *)
Table[n + d[n, 3, 2] + d[n, 5, 2],
  {n, 1, 50}]                        (* A206812 *)
Flatten[Table[p[2, n], {n, 1, z/8}]] (* A206813 *)
Table[n + d[n, 2, 3] + d[n, 5, 3],
  {n, 1, 50}]                        (* A206813 *)
Flatten[Table[p[3, n], {n, 1, z/8}]] (* A206814 *)
Table[n + d[n, 2, 5] + d[n, 3, 5],
  {n, 1, 50}]                        (* A206814 *)

PARI

a(n) = n + logint(2^n,3) + logint(2^n,5) \\ Ruud H.G. van Tol, Nov 16 2022

Formula

a(n) = n + [n*log_3(2)] + [n*log_5(2)],

A206813(n) = n + [n*log_2(3)] + [n*log_5(3)],

A206814(n) = n + [n*log_2(5)] + [n*log_3(5)],

where []=floor.