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%I #14 Sep 23 2020 03:03:54
%S 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,
%T 108,125,128,135,144,162,192,216,225,243,256,288,324,375,384,405,432,
%U 486,512,576,625,648,675,729,768,864,972,1024,1125,1152,1215,1296
%N Numbers of form (2^i)*(3^j) or (3^i)*(5^j).
%C Union of A003586 and A003593;
%C A006530(a(n)) <= 5; A001221(a(n)) <= 2; a(n) mod 10 != 0.
%H Reinhard Zumkeller, <a href="/A258023/b258023.txt">Table of n, a(n) for n = 1..10000</a>
%H Vaclav Kotesovec, <a href="/A258023/a258023.jpg">Graph - the asymptotic ratio (65000000 terms)</a>
%F a(n) ~ exp(sqrt(2*log(2)*log(3)*log(5)*n / log(10))) / sqrt(3). - _Vaclav Kotesovec_, Sep 22 2020
%F Sum_{n>=1} 1/a(n) = 27/8. - _Amiram Eldar_, Sep 23 2020
%e . n | a(n) | n | a(n) |
%e . ----+-------+---------- ----+-------+------------
%e . 1 | 1 | 1 16 | 32 | 2^5
%e . 2 | 2 | 2 17 | 36 | 2^2 * 3^2
%e . 3 | 3 | 3 18 | 45 | 3^2 * 5
%e . 4 | 4 | 2^2 19 | 48 | 2^4 * 3
%e . 5 | 5 | 5 20 | 54 | 2 * 3^3
%e . 6 | 6 | 2 * 3 21 | 64 | 2^6
%e . 7 | 8 | 2^3 22 | 72 | 2^3 * 3^2
%e . 8 | 9 | 3^2 23 | 75 | 3 * 5^2
%e . 9 | 12 | 2^2 * 3 24 | 81 | 3^4
%e . 10 | 15 | 3 * 5 25 | 96 | 2^5 * 3
%e . 11 | 16 | 2^4 26 | 108 | 2^2 * 3^3
%e . 12 | 18 | 2 * 3^2 27 | 125 | 5^3
%e . 13 | 24 | 2^3 * 3 28 | 128 | 2^7
%e . 14 | 25 | 5^2 29 | 135 | 3^3 * 5
%e . 15 | 27 | 3^3 30 | 144 | 2^4 * 3^2
%t 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 *)
%o (Haskell)
%o import Data.List.Ordered (union)
%o a258023 n = a258023_list !! (n-1)
%o a258023_list = union a003586_list a003593_list
%Y Cf. A003586, A003593, A051037, A006530, A001221, A010879, subsequence of: A051037, A257997, A337800, A337801.
%K nonn,easy
%O 1,2
%A _Reinhard Zumkeller_, May 16 2015
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