Below are the oldest existing lists of the bishops of Rome following Saint Peter. As you can see, Saint Linus (mentioned in 2 Tim 4:21) is universally held as the immediate successor of Saint Peter in Rome. Clement (Phil 4:3) is also reckoned as a successor.

• Linus, Cletus, Clemens (ca. A.D. 160, Hegesippus, ap. Epiphanium, Canon of Mass).
• Linus, Anencletus, Clemens (ca. A.D. 170s, Irenaeus, Africanus ap. Eusebium).
• Linus, Clemens, Cletus, Anacletus (220s-230s, Hippolytus).
• Linus, Cletus, Anacletus, Clemens (ca. late 200s [?], Poem against Marcion).
• Linus, Anacletus, Clemens (400s Jerome).
• Linus, Clemens, Anacletus (400s Optatus, Augustine).[1]

It seems that there is some confusion about “Anacletus.” In the Anti-Marcionite poem and in Hippolytus, the name became redundant and mistaken for two different bishops: “Cletus” and “Anacletus.” The older lists indicate that Cletus and Anacletus are one and the same man – the former being an abbreviation of the latter.

The received tradition is easy to remember. I teach my students to simply remember Rome as the “PLACE” of the first five Popes:

P – Peter
L – Linus
A – Anacletus
C – Clement
E – Evaristus

[1] Catholic Encyclopedia, “Linus” (can be accessed at newadvent.org).