OEIS link. This is the sequence of k so that

    \[\frac{k}{\lambda(k)}\]

is incrementally largest, where \lambda is the Carmichael function.

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

If we fix a list of primes, and write n=2^{a_0} p_1^{a_1}\dots p_k^{a_k} where only a_0 can be 0, we can find what the exponents should be. There are four cases for a_0=0,1,2 and a_0>2. I’ll do the last one and the others are analogous.

    \[\lambda(n)=\text{lcm}(2^{a_0-2},(p_1-1)p_1^{a_1-1},\dots,(p_k-1)p_k^{a_k-1})\]

and now we choose the a_i to be equal to or smaller than one (2 for a_0) plus the largest exponent of p_i which occurs in any of the other parentheses, because if we chose a_i to be greater than that it would have the effect of multiplying both n and \lambda(n) 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 n to be 2,3,5,7 we have that the possibilities of n are 105,210,315,420,630,840,1260,2520 and of those only 1260 appears on the sequence. One thought I had is that we would want n to be divisible by some prime p so that p-1 has a prime raised to a large power in its factorization, but for example n which is divisible by 2,17 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.