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A264164 3-smooth numbers whose number of divisors is not 3-smooth.

%I A264164 #21 Apr 17 2025 08:08:25
%S A264164 16,48,64,81,144,162,192,324,432,512,576,648,729,1024,1296,1458,1536,
%T A264164 1728,2592,2916,3072,3888,4096,4608,5184,5832,8192,9216,10368,11664,
%U A264164 12288,13824,15552,16384,19683,20736,23328,24576,27648,34992,36864,39366,41472
%N A264164 3-smooth numbers whose number of divisors is not 3-smooth.
%H A264164 Reinhard Zumkeller, <a href="/A264164/b264164.txt">Table of n, a(n) for n = 1..10000</a>
%F A264164 A065333(a(n)) * (1 - A065333(A000005(a(n)))) = 1.
%F A264164 Sum_{n>=1} 1/a(n) = 3 - (Sum_{n>=1} 1/2^A069353(n)) * (Sum_{n>=1} 1/3^A069353(n)) = 0.14870906483739045152... . - _Amiram Eldar_, Apr 17 2025
%e A264164 a(12) = 648 = 2^3*3^4 = A003586(36) and A000005(648) = 20 = 2^2*5.
%e A264164 a(13) = 729 = 3^6 = A003586(37) and A000005(729) = 7.
%e A264164 A003586(38) = 768 = 2^8*3 is not a term, as A000005(768) = 18 = 2*3^2.
%t A264164 smQ[n_] := n == Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]);
%t A264164 seq[max_] := Sort@ Flatten@ Table[If[smQ[i + 1] && smQ[j + 1], Nothing, 2^i * 3^j], {i, 0 , Log2[max]}, {j, 0, Log[3, max/2^i]}]; seq[42000] (* _Amiram Eldar_, Sep 03 2023 *)
%o A264164 (Haskell)
%o A264164 a264164 n = a264164_list !! (n-1)
%o A264164 a264164_list = filter ((== 0) . a065333 . a000005') a003586_list
%Y A264164 Cf. A000005, A003586, A065333, A069353, A264165 (complement with respect to A003586).
%K A264164 nonn
%O A264164 1,1
%A A264164 _Reinhard Zumkeller_, Nov 19 2015