A257997 Numbers of the form (2^i)*(3^j) or (2^i)*(5^j) or (3^i)*(5^j).

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 32, 36, 40, 45, 48, 50, 54, 64, 72, 75, 80, 81, 96, 100, 108, 125, 128, 135, 144, 160, 162, 192, 200, 216, 225, 243, 250, 256, 288, 320, 324, 375, 384, 400, 405, 432, 486, 500, 512, 576, 625, 640

Offset: 1

Author

Reinhard Zumkeller, May 16 2015

Keywords

• nonn
• easy

Comments

Union of A003586, A003592 and A003593.

Subsequence of 5-smooth numbers (cf. A051037), having no more than two distinct prime factors: A006530(a(n)) <= 5; A001221(a(n)) <= 2.

Examples

			. ----+------+---------     ----+------+-----------
.   1 |   1  |  1            16 |  25  |  5^2
.   2 |   2  |  2            17 |  27  |  3^3
.   3 |   3  |  3            18 |  32  |  2^5
.   4 |   4  |  2^2          19 |  36  |  2^2 * 3^2
.   5 |   5  |  5            20 |  40  |  2^3 * 5
.   6 |   6  |  2 * 3        21 |  45  |  3^2 * 5
.   7 |   8  |  2^3          22 |  48  |  2^4 * 3
.   8 |   9  |  3^2          23 |  50  |  2 * 5^2
.   9 |  10  |  2 * 5        24 |  54  |  2 * 3^3
.  10 |  12  |  2^2 * 3      25 |  64  |  2^6
.  11 |  15  |  3 * 5        26 |  72  |  2^3 * 3^2
.  12 |  16  |  2^4          27 |  75  |  3 * 5^2
.  13 |  18  |  2 * 3^2      28 |  80  |  2^4 * 5
.  14 |  20  |  2^2 * 5      29 |  81  |  3^4
.  15 |  24  |  2^3 * 3      30 |  96  |  2^5 * 3
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (80000000 terms)

Crossrefs

Cf. A003586, A003592, A003593, A051037, A006530, A001221, A258023 (subsequence).

Cf. A337800, A337801.

Programs

Haskell

import Data.List.Ordered (unionAll)
a257997 n = a257997_list !! (n-1)
a257997_list = unionAll [a003586_list, a003592_list, a003593_list]

Mathematica

n = 1000; 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]}], Table[2^i*5^j, {i, 0, Log[2, n]}, {j, 0, Log[5, n/2^i]}]] // Flatten // Union (* Amiram Eldar, Sep 23 2020 *)

Formula

a(n) ~ exp(sqrt(2*log(2)*log(3)*log(5)*n/log(30))). - Vaclav Kotesovec, Sep 22 2020

Sum_{n>=1} 1/a(n) = 29/8. - Amiram Eldar, Sep 23 2020