For the record (not an answer), the function $a(n-1,n)$ for $n$ up to $10^4$ contains 2264 distinct primes, the largest being equal to 20369. The growth rate of the primes is close to linear in $n$, see plot.

<IMG SRC="https://ilorentz.org/beenakker/MO/primes_growth.png" WIDTH="400"/>

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<sub> Mathematica code 

`out=Table[{t=RecurrenceTable[{a[m]==a[m-1]+GCD[a[m-1],n-m],a[0]==n},a,{m, 1,n-1}];{n,t[[n-1]],PrimeQ[t[[n-1]]]}},{n,2,10000}]; And @@ out[[All,1,3]]    
ListPlot[out[[All,1,2]]]`
</sub>