Numerical values of Latin letters

[Jonathan Post]

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."

Sum of the numerical equivalents for the 23 Latin letters,
according to Tartaglia, of the letters in the Latin name of n.

For n = 1, 2, 3, 4, ... we have:

170, 516, 560, 1421, 1351, 330, 2130, 282, 1356, 2100, 2195, 2616, ...

a(1) = A132475(UNUS) = A132475(V)+A322475(N)+A132475(V)+A322475(S) =
5+90+5+70 = 170.

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
>