This question is inspired by this post.
I wonder whether the number of squarefree integers $\ n\ge 2\ $ with $\ \varphi(n)\mid \sigma(n)\ $ is still infinite. As in the link , $\ \varphi(n)\ $ is the totient function and $\ \sigma(n)\ $ the divisor - sum function. The first $\ 40\ $ such numbers are : $$[2, 3, 6, 14, 15, 30, 35, 42, 70, 78, 105, 190, 210, 357, 418, 570, 714, 910, 1045, 1254, 2090, 2730, 3135, 4522, 4674, 5278, 6270, 10659, 12441, 13566, 14630, 15834, 16770, 20026, 21318, 23374, 24871, 24882, 24969, 25070]$$
I think that for a given number of prime factors, there are only finite many such numbers (can someone prove this ?) , but still infinite many such numbers could exist.
Below $\ 10^9\ $ there are $1126$ such numbers , the following table summarizes the number of numbers with $1,2,\cdots , 9$ prime factors
1 2
2 4
3 9
4 28
5 171
6 487
7 340
8 81
9 4
The $\ 183\ $- digit number $$148504621260915439590480104525880008534525196529172773528290153939780712483230238657438737509145961295133297108620156439175622984987119112903647214652622780573688542572639027308346330$$ gives ratio $15$ and has $40$ prime factors :
$$[2, 3, 5, 11, 19, 29, 31, 53, 97, 137, 269, 457, 757, 1741, 4987, 8731, 25013, 30211, 30631, 46511, 51797, 53101, 59921, 74729, 231223, 430949, 440399, 451249, 662029, 858001, 1061759, 7320541, 10086707, 11625827, 13551379, 13992817, 20885069, 23868931, 94944263, 4346121576069167]$$
There are $\ 941\ $ such numbers with no prime factor exceeding $\ 100\ $