[seqfan] Re: Verifying sequence pertaining to Roman numerals and divisors
Allan Wechsler
acwacw at gmail.com
Sat Mar 25 02:49:20 CET 2023
The Roman numeral system can be thought of as a core convention surrounded
by a nest of increasingly anomalous practices, shading off into outright
blunders. A famous historical artifact is a padlock thought to have been
used to lock the gates of Athlone, Ireland. It is stamped with a date, with
the year given as "XVIXIII" -- that is, 1613. Here we are clearly in the
borderlands of Roman numerical notation; I present this example as a
warning that things of the real world are seldom as well-defined as a
mathematician would wish
That said, the core convention is that a Roman numeral can be thought of as
a four-position structure, with the four positions representing thousands,
hundreds, tens, and units. In each position, the weights 1-9 are
represented by designators like the familiar I, II, III, IV, V, VI, VII,
VIII, IX, and 0 is represented by the null string. The core convention
peters out above 3,999; there is no clear consensus on how to represent
4,000, (I am *not* saying that nobody ever wrote 4,000 in Roman numerals;
I'm just saying that there are various ways to do it, and none of them are
clearly dominant.) Usages like "IIII", "XXXX", "DCCCC", are known, but
clearly peripheral, as are things like "IIX" for 8 and "IC" for 99. Yes,
you can probably find some manuscript that represented 995 as VM. No, such
practices were not common.
Assuming the convention being used is clearly included in the definition,
sequences like Al's proposal seem no crazier to me than dozens of other
sequences that refer to the decimal representation of integers.
On Fri, Mar 24, 2023 at 7:10 PM <hv at crypt.org> wrote:
> One possible issue is that there isn't a unique "Roman numeral
> representation" through history.
>
> If I remember correctly, for example, 4 was originally represented as
> IIII, and only later as IV. And in mediaeval times most of the alphabet
> was coopted for specialized uses.
>
> So this would probably need somehow to specify the particular form
> that was being considered.
>
> (I haven't checked whether there are existing sequences with this problem.)
>
> Hugo
>
> Alonso Del Arte <alonso.delarte at gmail.com> wrote:
> :Is this correct?
> :
> :Numbers *n* such that the Roman numeral representation has at least
> :one nontrivial divisor of *n* as a substring
> :
> :8, 12, 15, 18, 20, 22, 24, 25, etc.
> :
> :I've bolded some of the characters, but it may or may not go through:
> :
> :V*II*I, X*II*, X*V*, XV*II*I, *X*X, XX*II*, XX*IV*, XX*V*, etc.
> :
> :Not getting OEIS results, but I haven't tried variations of this yet.
> :
> :Al
> :
> :--
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