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When represented in base three, there are some integer powers of two where the digits are a permutation of the next power of two.
Some examples:
\$2^5\$ in base three is \$1012_3\$, whereas \$2^6\$ in base three is \$2101_3\$, both have one \$0\$, two \$1\$’s, and one \$2\$
\$2^{27}\$ → \$1001...