Prime Factor

A prime factor is a factor that is prime, i.e., one that cannot itself be factored. In general, a prime factorization takes the form

where are prime factors and are their orders. prime factorization can be performed in the Wolfram Language using the command FactorInteger[n], which returns a list of pairs.

The following table gives the prime factorization for the positive integers .

	prime fac.		prime fac.		prime fac.		prime fac.		prime fac.
1	1	11	11	21		31	31	41	41
2	2	12		22		32		42
3	3	13	13	23	23	33		43	43
4		14		24		34		44
5	5	15		25		35		45
6		16		26		36		46
7	7	17	17	27		37	37	47	47
8		18		28		38		48
9		19	19	29	29	39		49
10		20		30		40		50

Two standard arithmetic functions count prime factors. The prime nu function counts distinct prime factors, while the prime omega function counts them with multiplicity. For example, gives and (Hardy and Wright 1979, p. 354). The notation is also used for , while can denote either count, so care is needed unless the convention is stated. The appearing in Veritasium (2025) is instead a local notation for the number of Goldbach partitions of .