Wolfram Language (Mathematica), 8969 bytes
Takes a positive integer n and outputs all n-sociable sequences
Do[If[Last[s=NestList[Tr@Divisors@#Do[NestList[Tr@Divisors@#-#&,d,#]]==d&&Tr[1^Union@s]==##]/.{a__,Print@Most@s]d}/;0!=a:>Print@{a},{d,∞}]&
-20 bytes thanks to @att
Wolfram Language (Mathematica), 89 bytes
Takes a positive integer n and outputs all n-sociable sequences
Do[If[Last[s=NestList[Tr@Divisors@#-#&,d,#]]==d&&Tr[1^Union@s]==#,Print@Most@s],{d,∞}]&
Wolfram Language (Mathematica), 69 bytes
Takes a positive integer n and outputs all n-sociable sequences
Do[NestList[Tr@Divisors@#-#&,d,#]/.{a__,d}/;0!=a:>Print@{a},{d,∞}]&
-20 bytes thanks to @att
Wolfram Language (Mathematica), 89 bytes
Takes a positive integer n and outputs all n-sociable sequences
Do[If[Last[s=NestList[Tr@Divisors@#-#&,d,#]]==d&&Tr[1^Union@s]==#,Print@Most@s],{d,∞}]&