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A355580 Powerful 3-smooth numbers: numbers of the form 2^i * 3^j with i, j != 1.

%I A355580 #11 Jul 10 2022 03:56:33
%S A355580 1,4,8,9,16,27,32,36,64,72,81,108,128,144,216,243,256,288,324,432,512,
%T A355580 576,648,729,864,972,1024,1152,1296,1728,1944,2048,2187,2304,2592,
%U A355580 2916,3456,3888,4096,4608,5184,5832,6561,6912,7776,8192,8748,9216,10368,11664
%N A355580 Powerful 3-smooth numbers: numbers of the form 2^i * 3^j with i, j != 1.
%C A355580 This sequence is closed under multiplication.
%H A355580 Amiram Eldar, <a href="/A355580/b355580.txt">Table of n, a(n) for n = 1..10000</a>
%F A355580 Sum_{n>=1} 1/a(n) = 7/4.
%e A355580 a(2) = 4 = 2^2.
%e A355580 a(3) = 8 = 2^3.
%e A355580 a(8)= 36 = 2^2 * 3^2.
%t A355580 q[n_] := Module[{e = IntegerExponent[n, {2, 3}]}, e[[1]] != 1 && e[[2]] != 1 && Times@@({2, 3}^e) == n]; Select[Range[12000], q]
%o A355580 (PARI) is(n) = {my(f=factor(n)); n == 1 || (vecmax(f[,1]) <= 3 && vecmin(f[,2]) > 1)};
%o A355580 (Python)
%o A355580 from itertools import count, takewhile
%o A355580 def aupto(lim):
%o A355580     pows2 = list(takewhile(lambda x: x<lim, (2**i for i in count(2))))
%o A355580     pows3 = list(takewhile(lambda x: x<lim, (3**i for i in count(2))))
%o A355580     return sorted(c*d for c in [1]+pows2 for d in [1]+pows3 if c*d <= lim)
%o A355580 print(aupto(12000)) # _Michael S. Branicky_, Jul 08 2022
%Y A355580 Intersection of A001694 and A003586.
%Y A355580 Union of A151821, (A000244 \ {3}) and 36*A003586.
%Y A355580 A003586 \ (Union of A007283 and A008776).
%Y A355580 Cf. A355581.
%K A355580 nonn,easy
%O A355580 1,2
%A A355580 _Amiram Eldar_, Jul 08 2022