Suppose, $a$ and $b$ are coprime positive integers.
Dirichlet's theorem states that infinite many primes of the form $an+b$ exist, if $n$ runs over the positive integers.
Let $p_1,p_2,\cdots $ be the sequence of those primes in increasing order ($b$ does not count, if it is prime, because we assume $n>0$) and denote $$s_k:=\sum_{j=1}^k p_j$$
Now, define $f(a,b)=m$ as the smallest positive integer $m$ such that $s_m$ is a perfect square. If no such $m$ exists, then $f(a,b)$ should be undefined.
In other words, we sum up the primes of the form $an+b$ in increasing order and stop as soon as the sum is a perfect square. In this case , $f(a,b)$ is the number of primes we must sum up. If we never get a perfect square, $f(a,b)$ is not defined.
Can we decide for which pairs $(a/b)$ , $f(a,b)$ is defined ?
In the range $1\le a,b\le 10$, I noticed the following hard cases :
[8, 1]
[8, 7]
[8, 9]
[9, 1]
[9, 2]
[9, 7]
[9, 8]
[9, 10]
Is $f(a,b)$ defined for those pairs ?