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

Antti Karttunen antti.karttunen at gmail.com
Wed Aug 2 17:38:16 CEST 2023

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
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,


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