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.
