[seqfan] Does A029747 give _all_ the fixed points of Doudna sequence and its inverse A005941?

[Antti Karttunen]

Dear SeqFans,


How much I have ever played with Doudna-sequence (A005940) and its
variants, I still haven't been completely ensured about Reinhard
Zumkeller's  Aug 23 2006 comment, that
https://oeis.org/A029747 (Numbers of the form 2^k times 1, 3 or 5.)
give _all_ its fixed points (when interpreted as an offset-1 permutation).
Any further cases of {2^k times p}, where p is a prime > 5 are easily
excluded, but I just wonder, couldn't there be some mysterious odd
composite c that would also satisfy A005940(c) = c ?
(And thus also all numbers that are 2^k times c).
Note also that there are some near cases "in the divisibility sense",
see e.g.: https://oeis.org/A364551

Now, Reinhard didn't leave any note why A029747 should contain all the
fixed points, so maybe it was an obvious fact for him.

There are two related questions:
Does A007283 (3*2^n) give all the fixed points of map n -> A163511(n)?
(where A163511 is the mirror-image of Doudna),
and also, whether A335431 gives all fixed points of map n -> A332214(n)?

One (futile?) way of trying to prove the latter could be considering
how the value of function
https://oeis.org/A331410
changes after such mapping, but this will soon get very tedious,
especially when there are infinite number of such subcases to
consider. See https://oeis.org/A335879

And for A029747, we could consider on which n the inverse Möbius transform of
https://oeis.org/A364558 certainly cannot obtain zero values, but I'm
not sure whether that would close it, and in any case, I'm asking for
a simple proof.


Best regards,

Antti