OEIS link. This is the sequence of k so that is incrementally largest, where is the Carmichael function.

I wrote a math stackexchange post which I’ll reproduce below:

If we fix a list of primes, and write where only can be , we can find what the exponents should be. There are four cases for and . I’ll do the last one and the others are analogous. and now we choose the to be equal to or smaller than one (2 for ) plus the largest exponent of which occurs in any of the other parentheses, because if we chose to be greater than that it would have the effect of multiplying both and which would cancel out.

So this means that for any list of primes we know that it appears on the sequence a finite number of times and we can find what the exponents are. For example if we want the primes that divide to be we have that the possibilities of are and of those only appears on the sequence. One thought I had is that we would want to be divisible by some prime so that has a prime raised to a large power in its factorization, but for example n which is divisible by does not appear on the sequence. Larger Fermat primes don’t work either. Some of the terms in the sequence are primorials like for the totient function, but some don’t appear, for example 30 and 210 don’t.

The question of describing the numbers which appear on the sequence feels like it shouldn’t be hard, but I haven’t been able to prove anything about it, except the result above. It also seems like the sequence encodes something about the sequence of numbers one less than the primes, but it’s unclear what that is. Alas, a seemingly interesting problem I cannot solve.