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 some hints & remarks @ Ch by Ady TZIDON)

Is the operator symbol -- not the operation -- what is meant by an element of a given "set of mathematical operations"?
Given your inclusion of multifactorials and subfactorials, it may seem so. Of course, [ fact! ] is not a mathematical operator and may be used to imply any of the family of factorial operations that use the factorial symbol ! in its operator -- or does it include any of the factorial family members such as primorial and superfactorial?).
Is superscription (exponentiation, tetration) and subscription (base n, pentation) permitted? And would [ sqrt ] imply the radical sign and not the operation alone? If so, may it be concatenated with a superscripted integer n to form an nth root?
Does concatenation allow for expressions resolved before concatenating its result with other numbers or results of other expressions? 

Edited on September 11, 2020, 4:50 am

Posted by Dej Mar on 2020-09-11 03:22:28