Let's start with a triplet of integers, say (1, 2, 5) and a set of mathematical operations (+, -, *, /, ^, sqrt, fact!, concatenation, brackets).

Our task will be to represent all (or almost all - as explained below) integers from 1 to n using some or all of the initial triplet and any quantity of operations defined above.

So:
1=1
6=1+5
9=5*2-1
13=15-2
27=51-4!
60=12*5 etc

Let's define n as the first occurrence of not being able to find a valid representation for n+1 and for n+2. I believe that in our case n=17 (15+2), since neither 18 nor 19 get valid solutions.

You are requested to find a triplet of integers (a,b,c) enabling a maximal n.

(In reply to Record by Ady TZIDON)

{4, 4, 9} may be better than {4, 9, 9} as concatenating may not be that important, and 4 can easily be converted to a 9 by using the subfactorial: (!4). The numbers 49, 94, 449, 494, 944 can still be formed by concatenating. I do not see a need for 99, 499, 949, and 999.      

With factorials limited to the factorial, double factorial, triple factorial and subfactorial, I currently do not see how to progress beyond 196. That is, I have not been able to form 197 with the three digits {4, 4, 9} selected. There is an implied restriction to limit the multifactorials to an index less than or equal to 3, that is, quadruple factorials, quintuple factorials, sextuple factorials, etc. may not be allowed. If they were, the possible finite limit would definitely be higher.

Ady's clarification of what operations were permitted did seem to resolve most questions. As Charlie pointed out, brackets have been used more than for grouping, yet, as there are many different definitions that can be used for the brackets, it should be assumed that the bracket operator is limited to grouping. As no examples were given of superscription and subscription, and with the inclusion of the carat but no examples of multiple carats, there is implications that hyperoperations are limited to hyper-3 (exponentiation) or lower (multiplication and addition) and numbers and numbers used in expressions are limited to base 10 -- i.e., no factorial base or other bases other than base-10. 

I have yet to explore other digit combinations. For {4, 4, 9} it seems that n = 196.

Posted by Dej Mar on 2020-09-14 08:49:04