I'm rather interested in the
motivation underlying this problem, apart from the fact that one
solution is easy to find. And apparently another quite hard.
(Although I didn't use a number cruncher to be frank; I tested
for p,q <50, very small for this site's standards I admit).
It seems to me that the restrictions
don't allow for a lot of solutions. If we allow to loosen the
restrictions it still will be difficult to find large solutions.
The problem states:
I) S_p=s^r S_q
The first restriction will be hard
enough as it is.
For the second restriction:
S_p[n] = k^2 seems to give a
solution with p=3 for all n.
This is no surprise since
S_3[n]=S_1[n]^2; Nichomachus's theorem.
S_p will be of the form 7^k *
13^l * remaining term, k and l>=1.
[If n prime then S_p[n] is congruent
to sum(i,i=1..n) mod p, which gives S_p congruent to 7*13
mod p, so at least 0 mod 7 and 0 mod 13]
Testing for several p shows the
remaining term is not very likely to have the same factors,
where the biggest one grows rather fast. Furthermore the the
multiplicity of the factors 7 and 13 don't appear to have a
'regular' pattern (I missed it).
So we might be 'lucky' to find a
second solution with r=2, let alone a different r>2 prime.
If we alter the problem in the form:
S_p[s] = r S_q[s] with p,q,r,s prime
It's not that hard to find small
solutions for certain combinations of p,q;
It will be hard enough to find large
Even finding large solutions to t
S_p[s] = r S_q[s] with p,q,r,s,t prime will require some
Then again, I'm not a number
theorist, but as posed it looks a bit too much 'Wieferich' to