##!! WARNING !!## Given the above circumstances the answer below is completely wrong and should not be replicated rather than used for guideline. The problem lies in the fact that Assumptions DON'T affect summation indices hence they are disregarded in the code below.

You can evaluate the series using the following snippet:

Sum[1/(k n (k + n)^2), {k, 1, ∞}, {n, 1, ∞}, Assumptions -> {k ∈ Primes, n ∈ Primes}] 

Which evaluates to:

π^4/180 

!! WARNING !!

Given the above circumstances the answer below is completely wrong and should not be replicated rather than used for guideline. The problem lies in the fact that Assumptions DON'T affect summation indices hence they are disregarded in the code below.

You can evaluate the series using the following snippet:

Sum[1/(k n (k + n)^2), {k, 1, ∞}, {n, 1, ∞}, Assumptions -> {k ∈ Primes, n ∈ Primes}] 

Which evaluates to:

π^4/180 

You can evaluate the series using the following snippet:

Sum[1/(k n (k + n)^2), {k, 1, ∞}, {n, 1, ∞}, Assumptions -> {k ∈ Primes, n ∈ Primes}] 

Which evaluates to:

π^4/180 

##!! WARNING !!## Given the above circumstances the answer below is completely wrong and should not be replicated rather than used for guideline. The problem lies in the fact that Assumptions DON'T affect summation indices hence they are disregarded in the code below.

You can evaluate the series using the following snippet:

Sum[1/(k n (k + n)^2), {k, 1, ∞}, {n, 1, ∞}, Assumptions -> {k ∈ Primes, n ∈ Primes}] 

Which evaluates to:

π^4/180