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

1, 4, 8, 9, 16, 27, 32, 36, 64, 72, 81, 108, 128, 144, 216, 243, 256, 288, 324, 432, 512, 576, 648, 729, 864, 972, 1024, 1152, 1296, 1728, 1944, 2048, 2187, 2304, 2592, 2916, 3456, 3888, 4096, 4608, 5184, 5832, 6561, 6912, 7776, 8192, 8748, 9216, 10368, 11664

Offset

1,2

Comments

This sequence is closed under multiplication.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

Example

a(2) = 4 = 2^2.
a(3) = 8 = 2^3.
a(8)= 36 = 2^2 * 3^2.

Mathematica

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

Prog

(PARI) is(n) = {my(f=factor(n)); n == 1 || (vecmax(f[, 1]) <= 3 && vecmin(f[, 2]) > 1)};
(Python)
from itertools import count, takewhile
def aupto(lim):
    pows2 = list(takewhile(lambda x: x<lim, (2**i for i in count(2))))
    pows3 = list(takewhile(lambda x: x<lim, (3**i for i in count(2))))
    return sorted(c*d for c in [1]+pows2 for d in [1]+pows3 if c*d <= lim)
print(aupto(12000)) # Michael S. Branicky, Jul 08 2022

Crossrefs

Intersection of A001694 and A003586.

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

A003586 \ (Union of A007283 and A008776).

Cf. A355581.

Sequence in context: A067252 A348995 A324723 * A272758 A227645 A375160

Adjacent sequences: A355577 A355578 A355579 * A355581 A355582 A355583

Keyword

nonn,easy

Author

Amiram Eldar, Jul 08 2022

Status

approved