A258023 Numbers of form (2^i)*(3^j) or (3^i)*(5^j).

1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 16, 18, 24, 25, 27, 32, 36, 45, 48, 54, 64, 72, 75, 81, 96, 108, 125, 128, 135, 144, 162, 192, 216, 225, 243, 256, 288, 324, 375, 384, 405, 432, 486, 512, 576, 625, 648, 675, 729, 768, 864, 972, 1024, 1125, 1152, 1215, 1296

Offset: 1

Author

Reinhard Zumkeller, May 16 2015

Keywords

• nonn
• easy

Comments

Union of A003586 and A003593;

A006530(a(n)) <= 5; A001221(a(n)) <= 2; a(n) mod 10 != 0.

Examples

			.   n |  a(n) |                 n |  a(n) |
. ----+-------+----------     ----+-------+------------
.   1 |    1  |  1             16 |   32  |  2^5
.   2 |    2  |  2             17 |   36  |  2^2 * 3^2
.   3 |    3  |  3             18 |   45  |  3^2 * 5
.   4 |    4  |  2^2           19 |   48  |  2^4 * 3
.   5 |    5  |  5             20 |   54  |  2 * 3^3
.   6 |    6  |  2 * 3         21 |   64  |  2^6
.   7 |    8  |  2^3           22 |   72  |  2^3 * 3^2
.   8 |    9  |  3^2           23 |   75  |  3 * 5^2
.   9 |   12  |  2^2 * 3       24 |   81  |  3^4
.  10 |   15  |  3 * 5         25 |   96  |  2^5 * 3
.  11 |   16  |  2^4           26 |  108  |  2^2 * 3^3
.  12 |   18  |  2 * 3^2       27 |  125  |  5^3
.  13 |   24  |  2^3 * 3       28 |  128  |  2^7
.  14 |   25  |  5^2           29 |  135  |  3^3 * 5
.  15 |   27  |  3^3           30 |  144  |  2^4 * 3^2
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (65000000 terms)

Crossrefs

Cf. A003586, A003593, A051037, A006530, A001221, A010879, subsequence of: A051037, A257997, A337800, A337801.

Programs

Haskell

import Data.List.Ordered (union)
a258023 n = a258023_list !! (n-1)
a258023_list = union a003586_list a003593_list

Mathematica

n = 10^4; Join[Table[2^i*3^j, {i, 0, Log[2, n]}, {j, 0, Log[3, n/2^i]}], Table[3^i*5^j, {i, 0, Log[3, n]}, {j, 0, Log[5, n/3^i]}]] // Flatten // Union (* Amiram Eldar, Sep 23 2020 *)

Formula

a(n) ~ exp(sqrt(2*log(2)*log(3)*log(5)*n / log(10))) / sqrt(3). - Vaclav Kotesovec, Sep 22 2020

Sum_{n>=1} 1/a(n) = 27/8. - Amiram Eldar, Sep 23 2020