Inspired by this question.
For positive integer $m$ , let $\sigma(m)$ be the divisor-sum function.
Let $S$ be the set of positive integers $n$ satisfying $$\sigma(\sigma(n)+n)=2\sigma(n)$$ In the linked question, $n$ is assumed to be composite and it is asked whether there are still infinite many solutions.
For a prime number $n$ we have $$n\in S\iff 2n+1\ is\ prime$$ in other words a prime is in $S$ iff it is a Sophie-Germain prime.
Question $1$ : Has $S$ only one even entry ($n=2$) ? Upto $n=10^9$ , there are $1804$ composite solutions , all odd.
Question $2$ : If we demand $n$ to be non-squarefree that is there is a prime $p$ with $p^2\mid n$. Are there still infinite many numbers in $S$ ?
Upto $n=10^9$ , there are $91$ such solutions :
1 3887 [13, 2; 23, 1]
2 4235 [5, 1; 7, 1; 11, 2]
3 18515 [5, 1; 7, 1; 23, 2]
4 21413 [7, 2; 19, 1; 23, 1]
5 64325 [5, 2; 31, 1; 83, 1]
6 94259 [11, 2; 19, 1; 41, 1]
7 135641 [11, 2; 19, 1; 59, 1]
8 203203 [7, 2; 11, 1; 13, 1; 29, 1]
9 444125 [5, 3; 11, 1; 17, 1; 19, 1]
10 638911 [7, 2; 13, 1; 17, 1; 59, 1]
11 721039 [11, 2; 59, 1; 101, 1]
12 979825 [5, 2; 7, 1; 11, 1; 509, 1]
13 1303533 [3, 4; 7, 1; 11, 2; 19, 1]
14 2452307 [11, 2; 13, 1; 1559, 1]
15 2774893 [11, 2; 17, 1; 19, 1; 71, 1]
16 3986983 [7, 2; 11, 1; 13, 1; 569, 1]
17 4670827 [7, 2; 19, 1; 29, 1; 173, 1]
18 5937505 [5, 1; 7, 1; 17, 2; 587, 1]
19 6242845 [5, 1; 7, 2; 83, 1; 307, 1]
20 6524203 [7, 3; 23, 1; 827, 1]
21 7784775 [3, 3; 5, 2; 19, 1; 607, 1]
22 8165443 [11, 2; 13, 1; 29, 1; 179, 1]
23 9541571 [13, 3; 43, 1; 101, 1]
24 12855323 [13, 2; 29, 1; 43, 1; 61, 1]
25 15982075 [5, 2; 353, 1; 1811, 1]
26 17711785 [5, 1; 7, 2; 13, 1; 67, 1; 83, 1]
27 18333007 [7, 3; 11, 1; 43, 1; 113, 1]
28 25544189 [11, 2; 19, 1; 41, 1; 271, 1]
29 26411245 [5, 1; 7, 2; 23, 1; 43, 1; 109, 1]
30 29018339 [7, 2; 19, 1; 71, 1; 439, 1]
31 38293255 [5, 1; 7, 2; 11, 1; 13, 1; 1093, 1]
32 47075975 [5, 2; 17, 1; 257, 1; 431, 1]
33 47975075 [5, 2; 313, 1; 6131, 1]
34 55225825 [5, 2; 311, 1; 7103, 1]
35 56455399 [7, 3; 11, 1; 13, 1; 1151, 1]
36 57070525 [5, 2; 971, 1; 2351, 1]
37 57417041 [11, 2; 17, 1; 103, 1; 271, 1]
38 63920255 [5, 1; 7, 2; 17, 1; 103, 1; 149, 1]
39 76815775 [5, 2; 17, 1; 61, 1; 2963, 1]
40 77492393 [11, 2; 19, 1; 37, 1; 911, 1]
41 93993209 [19, 2; 31, 1; 37, 1; 227, 1]
42 94270463 [7, 5; 71, 1; 79, 1]
43 100735375 [5, 3; 13, 1; 61991, 1]
44 100786829 [11, 2; 13, 1; 17, 1; 3769, 1]
45 118049869 [7, 2; 19, 1; 23, 1; 37, 1; 149, 1]
46 121619825 [5, 2; 73, 1; 103, 1; 647, 1]
47 145022449 [11, 1; 13, 2; 181, 1; 431, 1]
48 147370685 [5, 1; 7, 2; 11, 1; 149, 1; 367, 1]
49 171094999 [23, 2; 281, 1; 1151, 1]
50 171254125 [5, 3; 7, 1; 19, 1; 10301, 1]
51 171667727 [7, 3; 11, 1; 173, 1; 263, 1]
52 196033409 [11, 1; 13, 2; 17, 1; 6203, 1]
53 205750571 [13, 2; 23, 1; 43, 1; 1231, 1]
54 237542689 [23, 2; 191, 1; 2351, 1]
55 246804943 [7, 1; 11, 1; 47, 2; 1451, 1]
56 252986195 [5, 1; 7, 1; 11, 2; 31, 1; 41, 1; 47, 1]
57 269806201 [7, 3; 17, 1; 46271, 1]
58 272101201 [17, 1; 23, 2; 79, 1; 383, 1]
59 294272605 [5, 1; 11, 2; 503, 1; 967, 1]
60 296686425 [3, 1; 5, 2; 7, 3; 19, 1; 607, 1]
61 317194493 [7, 2; 11, 1; 19, 1; 47, 1; 659, 1]
62 325791527 [23, 2; 193, 1; 3191, 1]
63 336256011 [3, 2; 7, 1; 13, 1; 151, 1; 2719, 1]
64 347093215 [5, 1; 7, 2; 79, 2; 227, 1]
65 362530025 [5, 2; 11, 1; 13, 1; 23, 1; 4409, 1]
66 393083027 [17, 1; 73, 2; 4339, 1]
67 400341427 [23, 1; 79, 2; 2789, 1]
68 401926175 [5, 2; 7, 2; 328103, 1]
69 413847707 [7, 1; 13, 2; 349829, 1]
70 414045835 [5, 1; 7, 2; 251, 1; 6733, 1]
71 428035685 [5, 1; 7, 1; 11, 2; 53, 1; 1907, 1]
72 485327899 [7, 2; 23, 1; 499, 1; 863, 1]
73 524725565 [5, 1; 7, 2; 13, 2; 19, 1; 23, 1; 29, 1]
74 539167409 [11, 2; 47, 1; 113, 1; 839, 1]
75 559493375 [5, 3; 7, 1; 17, 1; 29, 1; 1297, 1]
76 560044233 [3, 3; 7, 1; 31, 1; 61, 1; 1567, 1]
77 562128853 [11, 2; 13, 1; 191, 1; 1871, 1]
78 630265225 [5, 2; 17, 1; 71, 1; 20887, 1]
79 679338205 [5, 1; 7, 2; 13, 1; 263, 1; 811, 1]
80 691733735 [5, 1; 7, 2; 11, 1; 223, 1; 1151, 1]
81 692317087 [7, 1; 11, 1; 29, 2; 10691, 1]
82 693003227 [7, 2; 29, 1; 173, 1; 2819, 1]
83 711193351 [11, 2; 17, 1; 19, 1; 31, 1; 587, 1]
84 729085063 [7, 2; 31, 1; 89, 1; 5393, 1]
85 752731625 [5, 3; 107, 1; 167, 1; 337, 1]
86 892694803 [7, 1; 11, 2; 13, 1; 17, 1; 19, 1; 251, 1]
87 896928445 [5, 1; 7, 1; 61, 2; 71, 1; 97, 1]
88 942742535 [5, 1; 7, 1; 43, 1; 53, 2; 223, 1]
89 975607721 [11, 1; 23, 2; 389, 1; 431, 1]
90 978386893 [11, 1; 19, 2; 37, 1; 6659, 1]
91 994422871 [13, 2; 17, 1; 23, 1; 101, 1; 149, 1]
Question $3$ : If $\sigma(n)$ is odd , so $n=l^2$ or $n=2l^2$ for some positive integer $l$. Is $n=2$ the only solution with this restriction ?
Upto $n=10^{14}$ , the only solution of this form is $2$