Obviously there is a finite number of T(n) which are 47-smooth, which always piques my interest on where this upper bound is located at. After some research on Smooth numbers I found Størmer's theorem which actually explicitly shows a method to solve a problem by solving a lot of Pell's Equation, unfortunately this seems extremely hard Mathematically and Computationally.
At the bottom of Størmer's theorem I found a sequence of largest possible n's such that T(n) is pn-smooth which is the sequence A117581 on OEIS. I now have the bound I was looking for, 1109496723126.
All I did now was generate all 47-smooth numbers less than this bound using a new function added to my Essential Functions, k-smooth numbers, and then I just went through the list and checked if n and n+1 were both in this list.
One day I would like to come back and use Størmer's theorem properly to solve this!
Enter a number (yourinput)
Code will output the the sum of all indices n such that T(n) is yourinput-smooth
If you input the max = 41 give it 10-15 seconds