k=4, 6=5^1+1

k=5, 10=3^2+1

k=8, 28=3^3+1

k(k-1)/2=p^n+1

k^2-k-2p^n-2=0

quadratic formula

k=.5+.5sqrt(9+8p^n)

I just checked primes up to 100 and powers that kept the discriminant small enough to see whether the square root had a decimal on my calculator.  So there may be larger solutions, but if so they are pretty big (or n=1 and p>100).