Numerical values of Latin letters

[Maximilian Hasler]

On Nov 21, 2007 5:14 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
> It now becomes possible to take the clever advice of njas [20 Nov 07]:
> "Perhaps a more natural sequence would be to write the numbers
> one, two, three, ... in Latin, and then sum the values of the letters."
> We have to take David Garber's advice as well: "W=U=V, and J=I."'...)
> a(1) = A132475(UNUS) = A132475(V)+A322475(N)+A132475(V)+A322475(S) =
> 5+90+5+70 = 170.

At a first glance, I was tempted to reply the following:
"I'm sorry but I can't consider this sequence as very natural and/or useful..."

Also I'm wondering if it is well defined ; I think the writing out in
words might suffer from the same ill-definedness than the writing out
in symbols I,V,X,L,C,D,M ; I'm not sure if

> > quadraginta septem, duodequinquaginta, undequinquagginta, quinquaginta

is the only authoritative way ("quadraginta novem" impossible ?)
(I'm saying that since I was very recently surprised to find on
http://projecteuler.net/index.php?section=problems&id=89
the "rule" that "I" can only preceed "V" and "X" and not "L","C",...)

At a second glance, maybe one could do at least a bit of mathematics on it :
e.g., it might be self-similar in a relaxed sense (first differences
periodic except for indices {10k-1,10k} or so (maybe rather
{100k-1,100k} or worse, in view of oddities I alluded to).

M.H.


>
> a(2) = A132475(DUO) = A322475(D)+A132475(V)+A322475(O) = 500 + 5 + 11 = 516.
>
> a(3) = A132475(TRES) = A132475(T)+A322475(R)+A132475(E)+A322475(S) =
> 160+80+250+70 = 560.
>
> a(4) = Q+V+A+T+T+V+O+R = 500+5+500+160+160+5+11+80 = 1421.
>
> a(5) = Q+V+I+N+Q+V+E = 500+5+1+90+500+5+250 = 1351.
>
> a(6) = S+E+X = 70+250+10 = 330.
>
> a(7) = S+E+P+T+E+M = 70+250+400+160+250+1000 = 2130.
>
> a(8) = O+C+T+O = 11+100+160+11 = 282.
>
> a(9) = N+O+V+E+M = 90+11+5+250+1000 = 1356.
>
> a(10) = D+E+C+E+M = 500+250+100+250+1000 = 2100.
>
> a(11) = V+N+D+E+C+E+M = 5+90+500+250+100+250+1000 = 2195.
>
> a(12) = D+V+O+D+E+C+E+M = 500+5+11+500+250+100+250+1000 = 2616.
>
> Again, we can ask what is the fixed point of this mapping, what are
> the equivalence classes, what happens on iteration, and the like.
>
> I think that this would have made perfect sense to Tartaglia, and to a
> Roman two millennia ago.  Then Cardano would have published first...
>
> Happy Thanksgiving (to Americans),
>
> Best,
>
> Jonathan Vos Post
>
>
>
> On Nov 21, 2007 12:00 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
> > Continuing, if I'm right at all, we'd have a(1)-a(50) as
> >
> > A132984 4, 3, 4, 8, 7, 3, 6, 4, 5, 5, 7, 8, 8, 13, 9, 7, 11, 12, 11,
> >  7, 11, 10, 11, 15, 14, 10, 13, 13, 12, 8, 12, 11, 12, 16, 15, 11, 14,
> >  16, 15, 11, 15, 14, 15, 19, 18, 14, 17, 16, 12
> >
> > with 40 through 50 being:
> >
> > quadraginta, quadraginta unus, quadraginta duo, quadraginta tres,
> > quadraginta quattuor, quadraginta quinque, quadraginta sex,
> > quadraginta septem, duodequinquaginta, undequinquagginta, quinquaginta
> >
> > Now we have the makings of the inverse function:
> >
> > Least cardinal integer which has exactly n letters in its Latin
> > name [can someone please verify and extend?]
> >
> > n  a(n) namely
> > 3   2     duo
> > 4   1     unus
> > 5   9     novem
> > 6   7     septem
> > 7   5     quinque
> > 8   4     quattuor
> > 9   15   quindecim
> > 10 22   viginti duo
> > 11 17   septemdecim
> > 12 18   duodeviginti
> > 13 14   quattuordecim
> > 14 25   viginti quinque
> > 15 24   viginti quattuor
> > 16 34   triginta quattuor
> > 17 47   quadraginta septem
> > 18 45   quadraginta quinque
> > 19 44   quadraginta quattuor
> > 20 54   quinquaginta quattuor
> >
>