[seqfan] Re: Mapping problem
Don Reble
djr at nk.ca
Wed Nov 9 13:03:15 CET 2016
> Let S(n) be the largest subset of Z(n) fixed by the mapping n -> n^2, and
> let f(n) = |Z(n)|.
> For example, S(25) = {0, 1, 6, 11, 16, 21} is the largest set of residues
> modulo 25 fixed by the mapping n -> n^2, so f(25) = |S(25)| = 6.
> Can you find a formula for f(n) in terms of n?
It looks like f(n) is multiplicative.
f(2^n) = 2,
f(p^n) = 1 + (p^(n-1) * oddpart(p-1))
(p is an odd prime, oddpart(x) is the largest odd factor of x)
That makes sense, when you consider the multiplicative (modulo n)
group of numbers coprime to n.
If a cyclic subgroup has odd order, it is closed under duplication;
but if it has even order, it is not closed. (The generators aren't
duplicates of anything.)
That "1 +" in f(p^n) accounts for the 0 element of Z(n), which is
not in any multiplicative group.
Hmm... It could be the limit, as something goes to infinity,
of A000224, A052273, A085311, ...
--
Don Reble djr at nk.ca
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