I notice a pattern when computing the least common multiple for the numbers 1 through n as we incrementally increase n.
The pattern is that the list of factors does not increase unless the incremental next value we include is a prime number to a positive integer power. Also, if it does increase, it only ever increases by 1.
Consider the following python code. Notice that when we include 13, the factor list begins to include 13 because 13 is a prime raised to the first power. Notice that when we include 16, the factor list begins to include an additional 2 because 16 is a prime raised to the fourth power.
>>> factors(lcm(*range(1,13)))
[2, 2, 2, 3, 3, 5, 7, 11]
>>> factors(lcm(*range(1,14)))
[2, 2, 2, 3, 3, 5, 7, 11, 13]
>>> factors(lcm(*range(1,15)))
[2, 2, 2, 3, 3, 5, 7, 11, 13]
>>> factors(lcm(*range(1,16)))
[2, 2, 2, 3, 3, 5, 7, 11, 13]
>>> factors(lcm(*range(1,17)))
[2, 2, 2, 2, 3, 3, 5, 7, 11, 13]
>>> factors(lcm(*range(1,18)))
[2, 2, 2, 2, 3, 3, 5, 7, 11, 13, 17]
>>> factors(lcm(*range(1,19)))
[2, 2, 2, 2, 3, 3, 5, 7, 11, 13, 17]
>>> factors(lcm(*range(1,20)))
[2, 2, 2, 2, 3, 3, 5, 7, 11, 13, 17, 19]
Does the conjecture have a name, and is it true?
Stated in another way, we can always construct n+1 by rummaging around among the factors of the numbers 1 through n unless n+1 is a prime power. Further, if it is a prime power, the prime factorization of n+1 will always have just one additional factor than our previous budget had (e.g. 2^4 vs 2^3).
(for what it's worth, I was motivated to think about this because of a recent post from Fermat's Library: https://www.linkedin.com/posts/fermatslibrary_heres-a-surprising-appearance-of-e-the-activity-7101876060991950848-D7T-)