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

%I #11 Jul 10 2022 03:56:33

%S 1,4,8,9,16,27,32,36,64,72,81,108,128,144,216,243,256,288,324,432,512,

%T 576,648,729,864,972,1024,1152,1296,1728,1944,2048,2187,2304,2592,

%U 2916,3456,3888,4096,4608,5184,5832,6561,6912,7776,8192,8748,9216,10368,11664

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

%C This sequence is closed under multiplication.

%H Amiram Eldar, <a href="/A355580/b355580.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{n>=1} 1/a(n) = 7/4.

%e a(2) = 4 = 2^2.

%e a(3) = 8 = 2^3.

%e a(8)= 36 = 2^2 * 3^2.

%t q[n_] := Module[{e = IntegerExponent[n, {2, 3}]}, e[[1]] != 1 && e[[2]] != 1 && Times@@({2, 3}^e) == n]; Select[Range[12000], q]

%o (PARI) is(n) = {my(f=factor(n)); n == 1 || (vecmax(f[,1]) <= 3 && vecmin(f[,2]) > 1)};

%o (Python)

%o from itertools import count, takewhile

%o def aupto(lim):

%o pows2 = list(takewhile(lambda x: x<lim, (2**i for i in count(2))))

%o pows3 = list(takewhile(lambda x: x<lim, (3**i for i in count(2))))

%o return sorted(c*d for c in [1]+pows2 for d in [1]+pows3 if c*d <= lim)

%o print(aupto(12000)) # _Michael S. Branicky_, Jul 08 2022

%Y Intersection of A001694 and A003586.

%Y Union of A151821, (A000244 \ {3}) and 36*A003586.

%Y A003586 \ (Union of A007283 and A008776).

%Y Cf. A355581.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Jul 08 2022