The earliest list is Irenaeus, writing about 180 and quoted by Eusebius: Linus first after the apostles, then Anencletus, then Clement. Later Western tradition splits Anencletus into two men, Cletus and Anacletus, and Tertullian's claim that Peter ordained Clement himself pulls Clement forward in some reckonings. This dataset follows the Irenaean order because it is the earliest witness — which means the first ordinals on the Rome chain should be read as a later church's arithmetic, not as a record. Nobody was numbering bishops in the first century.
Ecclesiastical History · HE 3.2, 3.4.9, 3.13, 3.15 (quoting Irenaeus, Adv. Haer. 3.3.3)