Let $\lambda(m)$ be the Carmichael-function.
For which positive integers $n$ does a positive integer $m$ exist with $m+\lambda(m)=n$ ? In other words, for which $n$ is $m+\lambda(m)=n$ solvable ?
Of course this can be decided by checking all $m\le n$, but is there something better ? Or can we even classify the $n$ for which there is a solution ?
Some infinite families obviously have a solution $n=2^k\cdot 5$ , $k$ a positive integer (Solution $2^{k+2})$ or $p^{k-1}(2p-1)$ , $p^{k-1}(3p-1)$ , $p^{k-1}(5p-1)$ , $p$ an odd prime , $k$ a positive integer (solution : $p^k,2p^k,4p^k$ respectively) , but I cannot find a general pattern.
The first $50$ $n's$ with a solution are :
[2, 3, 5, 6, 8, 9, 10, 13, 14, 15, 19, 20, 21, 24, 25, 26, 27, 32, 33, 34, 37, 38, 40, 42, 43, 44, 45, 47, 48, 50, 51, 52, 54, 56, 57, 61, 62, 64, 67, 68, 69, 70, 72, 73, 75, 76, 77, 78, 80, 81]
Any ideas or references ?