A prime partition of a positive integer is a set of primes which sum to . For example, there are three prime partitions of 7 since

The number of prime partitions of , 3, ... are 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, ... (OEIS A000607). If for prime and for composite, then the Euler transform gives the number of partitions of into prime parts (Sloane and Plouffe 1995, p. 21).

The minimum number of primes needed to sum to , 3, ... are 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, ... (OEIS A051034). The maximum number of primes needed to sum to is just , 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ... (OEIS A004526), corresponding to a representation in terms of all 2s for an even number or one 3 and the rest 2s for an odd number.

The numbers which can be represented by a single prime are obviously the primes themselves. Composite numbers which can be represented as the
sum of two primes are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, ... (OEIS A051035), and composite numbers which are not
the sum of fewer than three primes are 27, 35, 51, 57, 65, 77, 87, 93, 95, 117, 119,
..., (OEIS A025583). The conjecture that *no*
numbers require four or more primes is called the Goldbach
conjecture.