Is there an odd $x$ such that $2x^2 \equiv 0 \pmod {\sigma(x)}$ and $\sigma(x^2) \equiv 0 \pmod {\sigma(x) - 1}$?

Question

CONTEXT

This question is a result of considerations stemming from this closely related MO question.

INITIAL QUESTION

My question is as is in the title:

Is there an odd $x$ such that $$2x^2 \equiv 0 \pmod {\sigma(x)}$$ and $$\sigma(x^2) \equiv 0 \pmod {\sigma(x) - 1}?$$

Here, $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$.

MY ATTEMPT

I tried searching for solutions to the congruences using GP in Sage Cell Server, I obtain:

For both congruences

for(x=2, 10000000, if((Mod(2*x^2,sigma(x)) == 0) && (Mod(sigma(x^2),sigma(x)-1) == 0),print(x,factor(x))))

No output returned.

for(x=2, 1000000, if((Mod(2*x^2,sigma(x)) == 0) && (Mod(sigma(x^2),sigma(x)-1) == 0),print(x,factor(x))))

No output returned.

For the congruence $2x^2 \equiv 0 \pmod {\sigma(x)}$

for(x=2, 10000000, if((Mod(2*x^2,sigma(x)) == 0),print(x,factor(x))))

Several lines returned. Will confirm if everything so far are even:

6[2, 1; 3, 1]
28[2, 2; 7, 1]
30[2, 1; 3, 1; 5, 1]
84[2, 2; 3, 1; 7, 1]
120[2, 3; 3, 1; 5, 1]
364[2, 2; 7, 1; 13, 1]
496[2, 4; 31, 1]
672[2, 5; 3, 1; 7, 1]
840[2, 3; 3, 1; 5, 1; 7, 1]
1080[2, 3; 3, 3; 5, 1]
1488[2, 4; 3, 1; 31, 1]
1782[2, 1; 3, 4; 11, 1]
2280[2, 3; 3, 1; 5, 1; 19, 1]
3276[2, 2; 3, 2; 7, 1; 13, 1]
3472[2, 4; 7, 1; 31, 1]
7440[2, 4; 3, 1; 5, 1; 31, 1]
7560[2, 3; 3, 3; 5, 1; 7, 1]
8128[2, 6; 127, 1]
8736[2, 5; 3, 1; 7, 1; 13, 1]
8910[2, 1; 3, 4; 5, 1; 11, 1]
9480[2, 3; 3, 1; 5, 1; 79, 1]
10920[2, 3; 3, 1; 5, 1; 7, 1; 13, 1]
11880[2, 3; 3, 3; 5, 1; 11, 1]
12400[2, 4; 5, 2; 31, 1]
16368[2, 4; 3, 1; 11, 1; 31, 1]
16380[2, 2; 3, 2; 5, 1; 7, 1; 13, 1]
18360[2, 3; 3, 3; 5, 1; 17, 1]
18600[2, 3; 3, 1; 5, 2; 31, 1]
20832[2, 5; 3, 1; 7, 1; 31, 1]
24384[2, 6; 3, 1; 127, 1]
24840[2, 3; 3, 3; 5, 1; 23, 1]
24948[2, 2; 3, 4; 7, 1; 11, 1]
25296[2, 4; 3, 1; 17, 1; 31, 1]
26208[2, 5; 3, 2; 7, 1; 13, 1]
30240[2, 5; 3, 3; 5, 1; 7, 1]
30256[2, 4; 31, 1; 61, 1]
30294[2, 1; 3, 4; 11, 1; 17, 1]
32760[2, 3; 3, 2; 5, 1; 7, 1; 13, 1]
35640[2, 3; 3, 4; 5, 1; 11, 1]
37200[2, 4; 3, 1; 5, 2; 31, 1]
45136[2, 4; 7, 1; 13, 1; 31, 1]
55692[2, 2; 3, 2; 7, 1; 13, 1; 17, 1]
55860[2, 2; 3, 1; 5, 1; 7, 2; 19, 1]
56896[2, 6; 7, 1; 127, 1]
57240[2, 3; 3, 3; 5, 1; 53, 1]
66960[2, 4; 3, 3; 5, 1; 31, 1]
76680[2, 3; 3, 3; 5, 1; 71, 1]
81480[2, 3; 3, 1; 5, 1; 7, 1; 97, 1]
84360[2, 3; 3, 1; 5, 1; 19, 1; 37, 1]
86800[2, 4; 5, 2; 7, 1; 31, 1]
90768[2, 4; 3, 1; 31, 1; 61, 1]
94446[2, 1; 3, 4; 11, 1; 53, 1]
115560[2, 3; 3, 3; 5, 1; 107, 1]
121920[2, 6; 3, 1; 5, 1; 127, 1]
122668[2, 2; 7, 1; 13, 1; 337, 1]
131040[2, 5; 3, 2; 5, 1; 7, 1; 13, 1]
149856[2, 5; 3, 1; 7, 1; 223, 1]
170688[2, 6; 3, 1; 7, 1; 127, 1]
173628[2, 2; 3, 2; 7, 1; 13, 1; 53, 1]
199584[2, 5; 3, 4; 7, 1; 11, 1]
201960[2, 3; 3, 3; 5, 1; 11, 1; 17, 1]
211792[2, 4; 7, 1; 31, 1; 61, 1]
215760[2, 4; 3, 1; 5, 1; 29, 1; 31, 1]
235600[2, 4; 5, 2; 19, 1; 31, 1]
249480[2, 3; 3, 4; 5, 1; 7, 1; 11, 1]
251968[2, 6; 31, 1; 127, 1]
268224[2, 6; 3, 1; 11, 1; 127, 1]
270816[2, 5; 3, 1; 7, 1; 13, 1; 31, 1]
288288[2, 5; 3, 2; 7, 1; 11, 1; 13, 1]
293760[2, 7; 3, 3; 5, 1; 17, 1]
332640[2, 5; 3, 3; 5, 1; 7, 1; 11, 1]
334800[2, 4; 3, 3; 5, 2; 31, 1]
336784[2, 4; 7, 1; 31, 1; 97, 1]
360360[2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]
393120[2, 5; 3, 3; 5, 1; 7, 1; 13, 1]
406224[2, 4; 3, 2; 7, 1; 13, 1; 31, 1]
409200[2, 4; 3, 1; 5, 2; 11, 1; 31, 1]
414528[2, 6; 3, 1; 17, 1; 127, 1]
441936[2, 4; 3, 4; 11, 1; 31, 1]
445536[2, 5; 3, 2; 7, 1; 13, 1; 17, 1]
453840[2, 4; 3, 1; 5, 1; 31, 1; 61, 1]
514080[2, 5; 3, 3; 5, 1; 7, 1; 17, 1]
520800[2, 5; 3, 1; 5, 2; 7, 1; 31, 1]
523776[2, 9; 3, 1; 11, 1; 31, 1]
556920[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 17, 1]
560832[2, 6; 3, 1; 23, 1; 127, 1]
574560[2, 5; 3, 3; 5, 1; 7, 1; 19, 1]
602784[2, 5; 3, 2; 7, 1; 13, 1; 23, 1]
605880[2, 3; 3, 4; 5, 1; 11, 1; 17, 1]
622440[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 19, 1]
629640[2, 3; 3, 3; 5, 1; 11, 1; 53, 1]
632400[2, 4; 3, 1; 5, 2; 17, 1; 31, 1]
677160[2, 3; 3, 4; 5, 1; 11, 1; 19, 1]
695520[2, 5; 3, 3; 5, 1; 7, 1; 23, 1]
698760[2, 3; 3, 3; 5, 1; 647, 1]
706800[2, 4; 3, 1; 5, 2; 19, 1; 31, 1]
739648[2, 6; 7, 1; 13, 1; 127, 1]
763308[2, 2; 3, 2; 7, 1; 13, 1; 233, 1]
812448[2, 5; 3, 2; 7, 1; 13, 1; 31, 1]
819000[2, 3; 3, 2; 5, 3; 7, 1; 13, 1]
819720[2, 3; 3, 4; 5, 1; 11, 1; 23, 1]
828816[2, 4; 3, 1; 31, 1; 557, 1]
853440[2, 6; 3, 1; 5, 1; 7, 1; 127, 1]
876960[2, 5; 3, 3; 5, 1; 7, 1; 29, 1]
884520[2, 3; 3, 5; 5, 1; 7, 1; 13, 1]
899808[2, 5; 3, 1; 7, 1; 13, 1; 103, 1]
950040[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 29, 1]
979600[2, 4; 5, 2; 31, 1; 79, 1]
997920[2, 5; 3, 4; 5, 1; 7, 1; 11, 1]
1033560[2, 3; 3, 4; 5, 1; 11, 1; 29, 1]
1078800[2, 4; 3, 1; 5, 2; 29, 1; 31, 1]
1097280[2, 6; 3, 3; 5, 1; 127, 1]
1108560[2, 4; 3, 1; 5, 1; 31, 1; 149, 1]
1128400[2, 4; 5, 2; 7, 1; 13, 1; 31, 1]
1146048[2, 6; 3, 1; 47, 1; 127, 1]
1230120[2, 3; 3, 3; 5, 1; 17, 1; 67, 1]
1231776[2, 5; 3, 2; 7, 1; 13, 1; 47, 1]
1239840[2, 5; 3, 3; 5, 1; 7, 1; 41, 1]
1270752[2, 5; 3, 1; 7, 1; 31, 1; 61, 1]
1341120[2, 6; 3, 1; 5, 1; 11, 1; 127, 1]
1343160[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 41, 1]
1421280[2, 5; 3, 3; 5, 1; 7, 1; 47, 1]
1441440[2, 5; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]
1503376[2, 4; 7, 1; 31, 1; 433, 1]
1543056[2, 4; 3, 1; 17, 1; 31, 1; 61, 1]
1602720[2, 5; 3, 3; 5, 1; 7, 1; 53, 1]
1675080[2, 3; 3, 4; 5, 1; 11, 1; 47, 1]
1731264[2, 6; 3, 1; 71, 1; 127, 1]
1784160[2, 5; 3, 3; 5, 1; 7, 1; 59, 1]
1854360[2, 3; 3, 3; 5, 1; 17, 1; 101, 1]
1860768[2, 5; 3, 2; 7, 1; 13, 1; 71, 1]
1888920[2, 3; 3, 4; 5, 1; 11, 1; 53, 1]
1932840[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 59, 1]
2056320[2, 7; 3, 3; 5, 1; 7, 1; 17, 1]
2102760[2, 3; 3, 4; 5, 1; 11, 1; 59, 1]
2147040[2, 5; 3, 3; 5, 1; 7, 1; 71, 1]
2178540[2, 2; 3, 2; 5, 1; 7, 2; 13, 1; 19, 1]
2194800[2, 4; 3, 1; 5, 2; 31, 1; 59, 1]
2209680[2, 4; 3, 4; 5, 1; 11, 1; 31, 1]
2218944[2, 6; 3, 1; 7, 1; 13, 1; 127, 1]
2296476[2, 2; 3, 2; 7, 1; 13, 1; 701, 1]
2316480[2, 6; 3, 1; 5, 1; 19, 1; 127, 1]
2340360[2, 3; 3, 3; 5, 1; 11, 1; 197, 1]
2388960[2, 5; 3, 3; 5, 1; 7, 1; 79, 1]
2467600[2, 4; 5, 2; 31, 1; 199, 1]
2489760[2, 5; 3, 2; 5, 1; 7, 1; 13, 1; 19, 1]
2509920[2, 5; 3, 3; 5, 1; 7, 1; 83, 1]
2530440[2, 3; 3, 4; 5, 1; 11, 1; 71, 1]
2594592[2, 5; 3, 4; 7, 1; 11, 1; 13, 1]
2691360[2, 5; 3, 3; 5, 1; 7, 1; 89, 1]
2699424[2, 5; 3, 2; 7, 1; 13, 1; 103, 1]
2719080[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 83, 1]
2804160[2, 6; 3, 1; 5, 1; 23, 1; 127, 1]
2815560[2, 3; 3, 4; 5, 1; 11, 1; 79, 1]
2863080[2, 3; 3, 3; 5, 1; 11, 1; 241, 1]
2901696[2, 6; 3, 1; 7, 1; 17, 1; 127, 1]
2915640[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 89, 1]
2944032[2, 5; 3, 1; 7, 1; 13, 1; 337, 1]
3013920[2, 5; 3, 2; 5, 1; 7, 1; 13, 1; 23, 1]
3171960[2, 3; 3, 4; 5, 1; 11, 1; 89, 1]
3231360[2, 7; 3, 3; 5, 1; 11, 1; 17, 1]
3235680[2, 5; 3, 3; 5, 1; 7, 1; 107, 1]
3243240[2, 3; 3, 4; 5, 1; 7, 1; 11, 1; 13, 1]
3276000[2, 5; 3, 2; 5, 3; 7, 1; 13, 1]
3310800[2, 4; 3, 1; 5, 2; 31, 1; 89, 1]
3328416[2, 5; 3, 2; 7, 1; 13, 1; 127, 1]
3392928[2, 5; 3, 4; 7, 1; 11, 1; 17, 1]
3535680[2, 6; 3, 1; 5, 1; 29, 1; 127, 1]
3666432[2, 9; 3, 1; 7, 1; 11, 1; 31, 1]
3800160[2, 5; 3, 2; 5, 1; 7, 1; 13, 1; 29, 1]
3813480[2, 3; 3, 4; 5, 1; 11, 1; 107, 1]
4021920[2, 5; 3, 3; 5, 1; 7, 2; 19, 1]
4062240[2, 5; 3, 2; 5, 1; 7, 1; 13, 1; 31, 1]
4203360[2, 5; 3, 3; 5, 1; 7, 1; 139, 1]
4241160[2, 3; 3, 4; 5, 1; 7, 1; 11, 1; 17, 1]
4253200[2, 4; 5, 2; 7, 3; 31, 1]
4324320[2, 5; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1]
4357080[2, 3; 3, 2; 5, 1; 7, 2; 13, 1; 19, 1]
4553640[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 139, 1]
4590432[2, 5; 3, 4; 7, 1; 11, 1; 23, 1]
4687200[2, 5; 3, 3; 5, 2; 7, 1; 31, 1]
5005728[2, 5; 3, 2; 7, 1; 13, 1; 191, 1]
5050080[2, 5; 3, 3; 5, 1; 7, 1; 167, 1]
5412960[2, 5; 3, 3; 5, 1; 7, 1; 179, 1]
5518912[2, 6; 7, 1; 97, 1; 127, 1]
5524200[2, 3; 3, 4; 5, 2; 11, 1; 31, 1]
5542800[2, 4; 3, 1; 5, 2; 31, 1; 149, 1]
5654880[2, 5; 3, 3; 5, 1; 7, 1; 11, 1; 17, 1]
5691600[2, 4; 3, 3; 5, 2; 17, 1; 31, 1]
5864040[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 179, 1]
5929560[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 181, 1]
6106464[2, 5; 3, 2; 7, 1; 13, 1; 233, 1]
6158280[2, 3; 3, 1; 5, 1; 19, 1; 37, 1; 73, 1]
6158880[2, 5; 3, 2; 5, 1; 7, 1; 13, 1; 47, 1]
6187104[2, 5; 3, 4; 7, 1; 11, 1; 31, 1]
6187600[2, 4; 5, 2; 31, 1; 499, 1]
6299200[2, 6; 5, 2; 31, 1; 127, 1]
6320160[2, 5; 3, 3; 5, 1; 7, 1; 11, 1; 19, 1]
6379560[2, 3; 3, 4; 5, 1; 11, 1; 179, 1]
6420960[2, 5; 3, 2; 5, 1; 7, 3; 13, 1]
6630624[2, 5; 3, 2; 7, 1; 11, 1; 13, 1; 23, 1]
6656832[2, 6; 3, 2; 7, 1; 13, 1; 127, 1]
6658800[2, 4; 3, 1; 5, 2; 31, 1; 179, 1]
6683040[2, 5; 3, 3; 5, 1; 7, 1; 13, 1; 17, 1]
6756480[2, 7; 3, 3; 5, 1; 17, 1; 23, 1]
6759060[2, 2; 3, 1; 5, 1; 7, 2; 11, 2; 19, 1]
6770400[2, 5; 3, 1; 5, 2; 7, 1; 13, 1; 31, 1]
6911760[2, 4; 3, 1; 5, 1; 31, 1; 929, 1]
6998208[2, 6; 3, 1; 7, 1; 41, 1; 127, 1]
7044840[2, 3; 3, 3; 5, 1; 11, 1; 593, 1]
7193280[2, 6; 3, 1; 5, 1; 59, 1; 127, 1]
7227360[2, 5; 3, 3; 5, 1; 7, 1; 239, 1]
7234920[2, 3; 3, 4; 5, 1; 7, 1; 11, 1; 29, 1]
7242048[2, 6; 3, 4; 11, 1; 127, 1]
7469280[2, 5; 3, 3; 5, 1; 7, 1; 13, 1; 19, 1]
7512912[2, 4; 3, 4; 11, 1; 17, 1; 31, 1]
7590240[2, 5; 3, 3; 5, 1; 7, 1; 251, 1]
7633080[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 233, 1]
7731360[2, 5; 3, 2; 5, 1; 7, 1; 13, 1; 59, 1]
8134560[2, 5; 3, 3; 5, 1; 7, 1; 269, 1]
8150688[2, 5; 3, 2; 7, 1; 13, 1; 311, 1]
8182944[2, 5; 3, 4; 7, 1; 11, 1; 41, 1]
8222760[2, 3; 3, 2; 5, 1; 7, 1; 13, 1; 251, 1]
8386560[2, 11; 3, 2; 5, 1; 7, 1; 13, 1]
8419600[2, 4; 5, 2; 7, 1; 31, 1; 97, 1]
8517960[2, 3; 3, 4; 5, 1; 11, 1; 239, 1]
8717200[2, 4; 5, 2; 19, 1; 31, 1; 37, 1]
8802624[2, 6; 3, 1; 19, 2; 127, 1]
8910720[2, 7; 3, 2; 5, 1; 7, 1; 13, 1; 17, 1]
8936928[2, 5; 3, 2; 7, 1; 11, 1; 13, 1; 31, 1]
9020256[2, 5; 3, 1; 7, 1; 31, 1; 433, 1]
9041760[2, 5; 3, 3; 5, 1; 7, 1; 13, 1; 23, 1]
9106560[2, 7; 3, 3; 5, 1; 17, 1; 31, 1]
9380448[2, 5; 3, 4; 7, 1; 11, 1; 47, 1]
9587160[2, 3; 3, 4; 5, 1; 11, 1; 269, 1]
9646560[2, 5; 3, 3; 5, 1; 7, 1; 11, 1; 29, 1]
9694080[2, 7; 3, 4; 5, 1; 11, 1; 17, 1]
9709200[2, 4; 3, 3; 5, 2; 29, 1; 31, 1]
9767520[2, 5; 3, 3; 5, 1; 7, 1; 17, 1; 19, 1]
9828000[2, 5; 3, 3; 5, 3; 7, 1; 13, 1]

for(x=2, 10000000, if((Mod(2*x^2,sigma(x)) == 0) && (Mod(x, 2) == 1),print(x,factor(x))))

No output returned. This confirms that the output for the preceding GP code are all even.

For the congruence $\sigma(x^2) \equiv 0 \pmod {\sigma(x) - 1}$

for(x=2, 1000000, if((Mod(sigma(x^2),sigma(x)-1) == 0),print(x,factor(x))))

52686[2, 1; 3, 2; 2927, 1]
54549[3, 2; 11, 1; 19, 1; 29, 1]
318528[2, 6; 3, 2; 7, 1; 79, 1]
520768[2, 6; 79, 1; 103, 1]
696045[3, 1; 5, 1; 7, 2; 947, 1]
925677[3, 2; 163, 1; 631, 1]

CONJECTURE

If one could show the truth of the following conjecture, then the problem in the hyperlinked MO question would be solved:

Conjecture: There is no odd $x > 1$ such that $2x^2 \equiv 0 \pmod {\sigma(x)}$.

FINAL QUESTION

Would you be able to prove the Conjecture?

Answer 1

I have not yet a proof, but some restrictions making it unlikely that $\sigma(x)\mid 2x^2$ is possible for an odd $x>1$

Assume there is such $x$. Then

• $x$ cannot be a prime power $p^k$ : $\sigma(p^k)=1+p+\cdots +p^k$ , hence $p\nmid \sigma(p^k)$. The only divisors of $2p^{2k}$ that are not divisible by $p$ are $1$ and $2$, but $\sigma(x)\ge 4$ is a divisor of $2x^2=2p^{2k}$

• Since $\sigma(n)$ cannot be divisible by $4$, at most one prime factor of $x$ has an odd exponent in the prime factorization. Hence $x$ must either be of the form $m^2$ or of the form $pm^2$ where $p$ is an odd prime and $m$ an odd integer. If the exponent is $1$, then the prime factor must be of the form $4k+1$

• If $p$ has exponent $1$ in the prime factorization of $x$, then $p+1\mid \sigma(x)\mid 2x^2$ hence every odd prime factor of $p+1$ must be a prime factor of $x$ and since $p+1$ cannot be divisible by $4$ but is at least $4$, $p+1$ actually must have an odd prime factor. In particular, the smallest prime factor cannot have exponent $1$.

• If $x$ has a prime factor smaller than $1\ 000$ , then $x>10^{12}$

• $x>10^9$ by brute force.