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Hansraj Gupta's Theorem 2 of his 1981 paper: the number vo of odd members of ϕ-1(2k) is 1 if 0 ≤ k < 32 and 0 for k ≥ 32. Furthermore, that paper includes an explicit list for m=576, as well as a useful table of even-odd checksums for ϕ-1([1,...,m]).
In Python testcases, the limit for time is 20 seconds and memory is 512 megabytes. My Python3 code runs under 360 milliseconds; its own prime factorization uses memo-ization.
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Project Euler #248: Numbers for which Euler’s totient function equals 13!
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To check the cardinality ("checksum") of your sets ϕ-1(m), consider these resources I used:
In Python testcases, the limit for time is 20 seconds and memory is 512 megabytes. My Python3 code runs under 360 milliseconds; its own prime factorization uses memo-ization.