Prime Partition

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.

REFERENCES:

Berndt, B. C. and Wilson, B. M. "Chapter 5 of Ramanujan's Second Notebook." In Analytic Number Theory: Proceedings of the Conference Held at Temple University, Philadelphia, Pa., May 12-15, 1980 (Ed. M. I. Knopp). Berlin: Springer-Verlag, pp. 49-78, 1981.

Chawla, L. M. and Shad, S. A. "On a Trio-Set of Partition Functions and Their Tables." J. Natural Sciences and Mathematics 9, 87-96, 1969.

Gupta, O. P. and Luthra, S. "Partitions into Primes." Proc. Nat. Inst. Sci. India. Part A 21, 181-184, 1955.

Gupta, H. "Partitions into Distinct Primes." Proc. Nat. Inst. Sci. India. Part A 21, 185-187, 1955.

Guy, R. K. "The Strong Law of Small Numbers." Amer. Math. Monthly 95, 697-712, 1988.

Sloane, N. J. A. Sequences A000607/M0265, A004526, A025583, A051034, and A051035 in "The On-Line Encyclopedia of Integer Sequences."

Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.