structural:summarization, information:classification
Some common primitive recursive functions

The following examples and definitions are from Kleene (1952) pp. 223–231. Many appear with proofs. Most also appear with similar names, either as proofs or as examples, in Boolos-Burgess-Jeffrey 2002 pp. 63–70; they add the logarithm lo(x, y) or lg(x, y) depending on the exact derivation.

In the following the mark " ' ", e.g. a', is the primitive mark meaning "the successor of", usually thought of as " +1", e.g. a +1 = a'. The functions 16–20 and #G are of particular interest with respect to converting primitive recursive predicates to, and extracting them from, their "arithmetical" form expressed as Gödel numbers.