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%I #23 Sep 23 2020 03:04:03
%S 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,
%T 72,75,80,81,96,100,108,125,128,135,144,160,162,192,200,216,225,243,
%U 250,256,288,320,324,375,384,400,405,432,486,500,512,576,625,640
%N Numbers of the form (2^i)*(3^j) or (2^i)*(5^j) or (3^i)*(5^j).
%C Union of A003586, A003592 and A003593.
%C Subsequence of 5-smooth numbers (cf. A051037), having no more than two distinct prime factors: A006530(a(n)) <= 5; A001221(a(n)) <= 2.
%H Reinhard Zumkeller, <a href="/A257997/b257997.txt">Table of n, a(n) for n = 1..10000</a>
%H Vaclav Kotesovec, <a href="/A257997/a257997_1.jpg">Graph - the asymptotic ratio (80000000 terms)</a>
%F a(n) ~ exp(sqrt(2*log(2)*log(3)*log(5)*n/log(30))). - _Vaclav Kotesovec_, Sep 22 2020
%F Sum_{n>=1} 1/a(n) = 29/8. - _Amiram Eldar_, Sep 23 2020
%e . ----+------+--------- ----+------+-----------
%e . 1 | 1 | 1 16 | 25 | 5^2
%e . 2 | 2 | 2 17 | 27 | 3^3
%e . 3 | 3 | 3 18 | 32 | 2^5
%e . 4 | 4 | 2^2 19 | 36 | 2^2 * 3^2
%e . 5 | 5 | 5 20 | 40 | 2^3 * 5
%e . 6 | 6 | 2 * 3 21 | 45 | 3^2 * 5
%e . 7 | 8 | 2^3 22 | 48 | 2^4 * 3
%e . 8 | 9 | 3^2 23 | 50 | 2 * 5^2
%e . 9 | 10 | 2 * 5 24 | 54 | 2 * 3^3
%e . 10 | 12 | 2^2 * 3 25 | 64 | 2^6
%e . 11 | 15 | 3 * 5 26 | 72 | 2^3 * 3^2
%e . 12 | 16 | 2^4 27 | 75 | 3 * 5^2
%e . 13 | 18 | 2 * 3^2 28 | 80 | 2^4 * 5
%e . 14 | 20 | 2^2 * 5 29 | 81 | 3^4
%e . 15 | 24 | 2^3 * 3 30 | 96 | 2^5 * 3
%t 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 *)
%o (Haskell)
%o import Data.List.Ordered (unionAll)
%o a257997 n = a257997_list !! (n-1)
%o a257997_list = unionAll [a003586_list, a003592_list, a003593_list]
%Y Cf. A003586, A003592, A003593, A051037, A006530, A001221, A258023 (subsequence).
%Y Cf. A337800, A337801.
%K nonn,easy
%O 1,2
%A _Reinhard Zumkeller_, May 16 2015
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