@mmcdara 
Thanks, there are not that many faculties numbers to investigate for a given number whether it is a faculties number 

Oh now I see you replied that in your opinion, no faculties numbers can be written symbolically in the series
Well, yes , I do recognise some of them now, so for a proof step well follow then? 

@sand15 
Thanks!, you are quite right that chatGPT is not a really solid foundation of knowledge, but I still find it very helpful to figure out some.
From orginele Euler pdf is quite a chore to get exactly clear why the sum: 1/n^2 (to infinity) = 1/6 Pi^2
Still try in Maple with the wiki examples

Let's look at sin(x) 
Now how to get n! numbers in the serie?

T:=(x,n)->subs(t=x,convert(taylor(sin(t),t=0,n+1),polynom));
   T := proc (x, n) options operator, arrow; subs(t = x, 
      convert(taylor(sin(t), t = 0, n+1), polynom)) end proc


sin(x)=T(x,10)+`...`;
                  1  3    1   5    1    7     1     9      
     sin(x) = x - - x  + --- x  - ---- x  + ------ x  + ...
                  6      120      5040      362880         

@vv 
Thanks, 
"but note that Maple cannot use ithprime symbolically."

I don't think it's a problem, because from comparing 2 finite sequences, Euler could derive something  
So first get the two series according to me.
From the euler product formula you have given, a partial sum can be made 

"1. primes (wrongly suposed to be the set of prime numbers in Maple) is infinite, so it cannot work for Product. Maple rejects even the syntax.
This is euler product formule ....
From this defintion of the zeta function as an euler -produkt formula, this maple code cannot make out the series... should be possible I suppose
Possibly the hypergeometric function is related to the eulerprodukt formula ?

Interesting all those examples on Wikipedia with proof steps.
The question is whether this can also be done in Maple, without too much complicated maths ?
Anyway, the AI generated some maple code and to what extent this can be used ?

@vv 

Thanks, I see in the Maple examples that a closed form for a sequence can also be found and then the convergence of it can be determined
yes, indeed only convergence can be determined from certain sums, apparently. 
If it becomes too difficult to represent an outcome in steps.
If convergence is established, then you can try to solve it further.
The code is generated by AI and I can look at multiple variants in Maple and see step by step where it is "going wrong" and try to find a solution in Maple?

I can try to go further with this code and try to solve the steps 
Note that  coeff is a protected name in Maple 

A plan for solving

@Preben Alsholm 
 

Thanks, yes maybe if it's just about the integral curve itself 
Seeing the line element field I also find illustrative.
Maybe DEplot cannot provide animation ?
There is a separate dfieldplot command and when using DEplot it does provide a line element field.
Understand that with numerically solving an ode, there is always a solution to be found ?
Are nice examples to keep as procedures.

@Preben Alsholm 
 

Thanks, these two code examples, are more advanced and the last one of them the concept is not entirely clear to me.
( it is an operation on a list L to get it in a different shape )
The steps I see here
                   L := [cos(t), -sin(t)]

                        {cos(t), sin(t)}

                       XY2 := [cos, -sin]

Should have more experience with it.
Apparently you can't remove (t) from list L immediately with the next step ?

@Preben Alsholm 
 

Thanks, I first created the procedure myself for the spiral field plot.
Then I asked AI (chatGPT) to create the procedure and that is now the code you are working on ..it should be a spiral via DEplot.

It's interesting when you (re)ask the AI to give as many variants as possible of Maple code for the task of showing two differential equations with two initial conditions in a field plot.
The AI never comes up with error-free code,but if you go through the code step by step, you will soon have a start.
In Maple, there are more roads that lead to Rome ( Dutch saying)
 

@Preben Alsholm 
Thanks, the idea is now further, how to make a function of x out of a list [a,b] 
note: keeping a set {a,b}  is probably not working for ~ operator

@janhardo 
If you then look at the AI generated code and step by step the the commands are already wrong .
Maybe in outline right ? 

# Definieer de differentiaalvergelijkingen
eq1 := diff(x(t), t) = y(t);
eq2 := diff(y(t), t) = -x(t);

# Geef de beginwaarden op
beginwaarden := x(0) = 1, y(0) = 0;
                             d             
                     eq1 := --- x(t) = y(t)
                             dt            

                            d              
                    eq2 := --- y(t) = -x(t)
                            dt             

               beginwaarden := x(0) = 1, y(0) = 0

# x := unapply(rhs(dsolve({eq1, eq2, beginwaarden}, {x(t), y(t)})), t);
dsolve({eq1, eq2, beginwaarden}, {x(t), y(t)});
                {x(t) = cos(t), y(t) = -sin(t)}

rhs(dsolve({eq1, eq2, beginwaarden}, {x(t), y(t)}));

? rhs
Error, invalid input: rhs received {x(t) = cos(t), y(t) = -sin(t)}, which is not valid for its 1st argument, expr

To pick the rightside from dsolve :

{x(t) = cos(t), y(t) = -sin(t)}

with rhs in order to make a function of x for cos(t)? : {x(t) = cos(t), y(t) = -sin(t)} 
Can be a sparringpartner the AI for problem solving , if the generated code is useful? 
Well, i stop with this getting the code working
note: i removed the t in the own made procedure: dv_systeemplot and it is working.

@acer 
Thanks, Ah, of course at Questions I must be ....
Want to make the procedure a bit more user-friendly, that would be next.
Will come back to a question on a new topic in due course.

@mmcdara 
Thanks, seems simpler now the procedure without using local variables.
The question comes up, when are local variables defined in a procedure ? 
@acer has already suggested something about this, but what he means exactly is not yet clear to me.