[seqfan] Re: Brief names for A234741 and A234742 wanted.
antti.karttunen at gmail.com
Tue Feb 18 04:29:55 CET 2014
On Mon, Feb 17, 2014 at 2:08 AM, M. F. Hasler <oeis at hasler.fr> wrote:
> From: M. F. Hasler <oeis at hasler.fr>
> Date: Sun, Feb 16, 2014 at 11:57 AM
> Subject: Re: [seqfan] Re: Brief names for A234741 and A234742 wanted.
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Dear Antti & SeqFans,
> I suggest to go slowly with inventing new (potentially "obscure")
> 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
> (These are just suggestions in reply to the request ; I don't want at all to
> impose my personal ideas.)
My comments to above.
First of all, what does "straightforward product" mean? (Ordinary
Then... Neologisms are also quite common in peer-reviewed journals. See e.g.:
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.
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 of my
old for A234742:
"a(n) = Number obtained when the binary encodings of irreducible
polynomial factors of GF(2)[X]-polynomial whose encoding n is, are
multiplied together normally, as natural numbers, with carry-bits
having their effect."
And likewise for A234741:
"Encoding of the product of the GF(2)[X]-polynomials represented by
the prime factors of n, with multiplicity."
is again more brief than my original
"a(n) = Number obtained when the prime divisors of n are multiplied
together as (encodings of) GF(2)[X]-polynomials (without carry-bits,
as in A048720)."
and includes a slight change of viewpoint as well.
Still, if we want to expand the current name of
A236380 a(n) = A234742(n) - A234741(n).
using these, it grows too long:
"Difference of product of the binary encodings of the irreducible
factors of the GF(2)[X]-polynomial whose encoding is n, with
multiplicity, and the encoding of the product of the
GF(2)[X]-polynomials represented by the prime factors of n, with
and essentially, it easily obscures the mathematical reciprocity that
is present in the definitions of A234741 and A234742.
For this reason, I'm still advocating for my neologism pairs "downward
remultiplication" and "upward remultiplication".
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.
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].
(Why I consider the latter be more primitive than the former? For
example, the latter (would) require less hardware to implement in
computer circuitry, because of the absence of carry logic. But this is
a viewpoint of an autodidact drop-out hacker. I wonder what a real
algebraicist would say here?)
We got also an added bonus of mnemonic value with these metaphoras:
Because of "monotone" effect of the carries, with downward
remultiplication Adownward(n) from Z to any of GF(p)[X], we have
Adownward(n) <= n, and likewise, with upward remultiplication
Aupward(n) from any of GF(p)[X] to Z, we have Aupward(n) >= n, so
these names also remind of other behaviour of these functions.
(See e.g. https://oeis.org/A234741/graph )
So, to use exactly these terms, one would say that
A234741(n) = Downward remultiplication from Z to GF(2)[X].
A234742(n) = Upward remultiplication from GF(2)[X] to Z.
(with the more traditional technical definitions moved either to the
Comments-section or after a semicolon on the Name-line).
and when the context is clear (e.g. on the Comments-section), one can
leave the unpronounceable part "GF(2)[X] to Z" off, and just talk
about "upward remultiplication".
Then the new name of https://oeis.org/draft/A236380
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].
Also, then I would have a coherent naming system for twenty something
new sequences that relate to A234741 and A234742 (which also come in
pairs, again "reciprocally related to each other", mutatis mutandis),
so we would have e.g.:
A236842 = "Numbers which can be expressed as a result of upward
remultiplication of some integer (from GF(2)[X] to Z)" instead of
just "A234742 sorted and duplicates removed."
A236834 = "Numbers which cannot be expressed as a result of downward
remultiplication (from Z to GF(2)[X]) of any number."
instead of current "Numbers that do not occur in A234741; positions of
zeros in A236833."
Do others have any opinions regarding this?
> On Sat, Feb 15, 2014 at 6:28 PM, Antti Karttunen <antti.karttunen at gmail.com>
>> On Sat, Feb 15, 2014 at 10:29 PM, Antti Karttunen
>> <antti.karttunen at gmail.com> wrote:
>> > Cheers,
>> > For several reasons I would like to have brief nicknames for these two
>> > sequences:
>> > http://oeis.org/A234741 a(n) = Number obtained when the prime divisors
>> > of n are multiplied together as (encodings of) GF(2)[X]-polynomials
>> > (without carry-bits, as in A048720).
>> > and:
>> > http://oeis.org/A234742 a(n) = Number obtained when the binary
>> > encodings of irreducible polynomial factors of GF(2)[X]-polynomial
>> > whose encoding n is, are multiplied together normally, as natural
>> > numbers, with carry-bits having their effect.
>> > I have been thinking about neologisms such as "downcarrying" for
>> > A234741 and "upcarrying" for A234742. But maybe they should also
>> > reflect that process akin of "unfolding open" and then "folding it
>> > again", but by "different creases"?
>> > (You may think that these sequences are about
>> > "unlawfully" mixing division & multiplication operations in two
>> > different rings, Z and GF(2)[X] polynomials)
>> > Also wondering whether Marc LeBrun's "rebasing" terminology would fit
>> > here somehow?
>> > (E.g. A234741 = "Number rebased to GF(2)[X] by its prime divisors" ?
>> > No...)
>> > Any suggestions are welcome.
>> I think I found it:
>> I will talk about "downward remultiplication (Z -> GF(2)[X])" and
>> "upward remultiplication (GF(2)[X] -> Z)" of numbers.
>> I realized also funny etymological truth: multiply = multi+ply, where
>> ply ~ fold ~ bend. At least according to Wiktionary, which I admit,
>> sometimes lies:
>> Then the sorted version of A234742 (with duplicates removed)
>> would have a name like:
>> "Numbers which occur as results of upward remultiplication (GF(2)[X]
>> -> Z, A234742) of some number."
>> And a sequence like:
>> would be:
>> "Numbers which do not occur as a result of downward remultiplication
>> (Z -> GF(2)[X], A234741) of any number."
>> Any objections or better ideas?
>> > Thanks,
>> > Antti
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