An Interesting Sequence

Zanka finds fun in everyday simple things. One fine day he got interested in a very trivial sequence. Given a natural number $k$, he considers the sequence $A_i=k+i^2$ so that the terms are $$k+1,k+4,k+9,k+16, \ldots $$ Now to make things more interesting, he considers the $\gcd$ of consecutive terms in the sequence, then takes the sum of the first $2k$ values. More formally, he wants to co