The motivation comes from the following question on MathOverflow:
Is the exponential version of Catalan-Dickson conjecture true?
Question 1. Is there a natural number $n$ satisfying the equation $n(n+1)=2^{[\log^{n}_{2}]+2}$ where $[.]$ indicates the floor function. I am also interested in the minimum $n$ satisfying this equation if there is any.
As the answer to the above question is negative, let's consider the following more generalized form:
Question 2. Are there natural number $n$ and prime number $p$ satisfying the equation $\frac{n(n+1)}{2}=\frac{p}{p-1}\times p^{[\log_{p}^{n}]}+\frac{p-2}{p-1}\times p^n$ where $[.]$ indicates the floor function. (Note that for $p=2$ we get the previous equation.)
The RHS of $$n(n+1)=2^{[\log^{n}_{2}]+2}$$ is a power of $2$ while the LHS is not a power of $2$ unless $n=1$ and $n=1$ is not a solution.
Thus I would say there are no solutions for this equation.