[seqfan] Re: Brief names for A234741 and A234742 wanted.

[M. F. Hasler]

Dear Antti, SeqFans,
apologies in advance for another lengthy reply...
to spoil the fun and make it short, my suggestion is:

A234741(n) = GF2X_to_N( remultiply( N_to_GF2X( factor( n ))))

or maybe "(GF(2)[X] -> N)" instead of "GF2X_to_N".
Details follow ; those not interested may hit "delete" now :-)


On Mon, Feb 17, 2014 at 11:29 PM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> On Mon, Feb 17, 2014 at 2:08 AM, M. F. Hasler <oeis at hasler.fr> wrote:
> > Dear Antti & SeqFans,
> >
> > I suggest to go slowly with inventing new (potentially "obscure")
> > terminology, at least it should IMHO not be used in the %NAME,
> > but maybe rather suggested in the comments.
> > I proposed to simplify %N A234742 to
> >
> > Product of the binary encodings of the irreducible factors (with
> > multiplicity) of the GF(2)[X]-polynomial whose encoding is n.
> >
> > and added the explaining comment:
> > The product is the straightforward product of integers.
> >
>
> > see https://oeis.org/draft/A234742
>
> First of all, what does "straightforward product" mean? (Ordinary
> multiplication?)

of course yes, ordinary multiplication...

> Then... Neologisms are also quite common in peer-reviewed journals.

I am not so much against Neologisms defined e.g. in comments, but IMHO
the supreme goal of the "NAME" is to make clear what the sequence is
about when you see nothing else of it.

> Although some of my own I have used in OEIS have not been so well
> crafted, (and I'm going to kill at least one of them completely), I
> still want to defend my "downward and upward remultiplication" for
> A234741 & A234742.

That may be fine, but we should try to use the KISS principle and
utmost elementary notions as far as possible in the %N line, I think.

> First, I admit that your version of the name: "Product of the binary
> encodings of the irreducible factors of the GF(2)[X]-polynomial whose
> encoding is n, with multiplicity." is a considerable improvement (...)
> Still, if we want to expand the current name of
> A236380 a(n) = A234742(n) - A234741(n).
> using these, it grows too long:
(I agree)

> For this reason, I'm still advocating for my neologism pairs "downward
> remultiplication" and "upward remultiplication".

When we use unknown words, then we could also leave the A-numbers...
But if we find something short and meaningful, then I'd agree.
[And we should find some meaningful description of this quite
"fundamental" operation.]
I frown upon "remultiplication" ("re-" means to do the same thing twice).

There is the aspect of factorization and "factorback" (in PARI speak),
[*]  but there is also the aspect of "encoding" which is essential :
natural numbers N <-> binary representation <-> GF(2)[X]

[*edit: now occurs to me that we may be saying exactly the same but
you just use "remultiplication" for what I call "factorback"... and I
admit that "remultiply" may be better than "factorback"!]


Actually, your sequences are something like
a(n) = GF2X_to_N( factorback( N_to_GF2X( factor( n ))))
and
b(n) = factorback( GF2X_to_N( factor( N_to_GF2X( n ))))

unless I didn't get it right.

I don't know whether these names are well chosen, but I think that the
unique(?) "canonical"  or at least most simple / natural(?) meaning of
the bijection
N_to_GF2X(.)  and its inverse GF2X_to_N(.)
can be found / guessed (more easily than the rules of chess) after
some thinking,...



> The informal definition for these is that in upward remultiplication,
> we are taking the irreducible factors obtained when n is considered as
> representing e.g. a polynomial in the polynomial ring GF(2)[X],
> GF(3)[X], etc. (via its base-2, base-3, etc. representation), and
> using the integers which encode those factors (in base-2, base-3, etc.
> respectively) to remultiply them in "higher" ring, for example, we can
> go from GF(2)[X] to GF(3)[X], and from GF(3)[X] to GF(5)[X] or to Z.

well I think /that/ is quite a bit different from the *bijection* between
N  and GF(2)[X].


> In downward remultiplication we do the opposite: We go from a
> "higher" ring to a "more primitive", e.g. from Z to GF(2)[X].

I think it's unneeded to impose an a priori idea of what is higher
resp. more primitive ; just say from where to where it goes.
The  injection GF(n) -> GF(m) (not a morphism), induces the same on
the ring of polynomials (just acting on coefficients) -- did I
understand that right?
Or do you use the bijection N <-> GF(n)[X], considering a polynomial
as the base-n expansion of a number?
(Then of course we have bijections, but still not morphisms for
addition or mulitplication. Is this the "carry bit" thing?)


> A234741(n) = Downward remultiplication from Z to GF(2)[X].

Are you sure you mean Z (with signs) rather than N (without signs)?
Anyway, I think it's not possible to guess what "downward" means;
OTOH:
A234741(n) = GF2X_to_N( remultiply( N_to_GF2X( factor( n ))))
/could/ be guessed, IMHO.
(To some of us, "(GF(2)[X] -> N)" might be preferrable to "GF2X_to_N".)



> Then the new name of https://oeis.org/draft/A236380 would be:
> a(n) = A234742(n) - A234741(n): Difference between the results when n
> is remultiplied upwards from GF(2)[X] to Z, and when it is
> remultiplied downwards from Z to GF(2)[X].

"the results when..is" can be deleted:

a(n) = A234742(n) - A234741(n): Difference between n remultiplied
upwards from GF(2)[X] to Z, and
remultiplied downwards from Z to GF(2)[X].


> Do others have any opinions regarding this?

remains an open question ;-)

Best regards,
Maximilian