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A factorial x! is the product of all numbers from 1 to x. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). 0! = 1 is sometimes included. A k-smooth number (for a natural number k) has its prime factors ≤ k (so it is also j-smooth for any j > k).
If 2 k + 1 is prime and k > 0, then k itself must be a power of 2, [1] so 2 k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023 [update] , the only known Fermat primes are F 0 = 3 , F 1 = 5 , F 2 = 17 , F 3 = 257 , and F 4 = 65537 (sequence A019434 in the OEIS ).
a prime number has only 1 and itself as divisors; that is, d(n) = 2 a composite number has more than just 1 and itself as divisors; that is, d ( n ) > 2 a highly composite number has a number of positive divisors that is greater than any lesser number; that is, d ( n ) > d ( m ) for every positive integer m < n .
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
If p is an irregular prime and p divides the numerator of the Bernoulli number B 2k for 0 < 2k < p − 1, then (p, 2k) is called an irregular pair. In other words, an irregular pair is a bookkeeping device to record, for an irregular prime p , the particular indices of the Bernoulli numbers at which regularity fails.
As 120 is a factorial and one less than a square (! =), it—with 11—is one of the few Brown number pairs. 120 appears in Pierre de Fermat's modified Diophantine problem as the largest known integer of the sequence 1, 3, 8, 120. Fermat wanted to find another positive integer that, when multiplied by any of the other numbers in the sequence ...
In number theory, the prime omega functions and () count the number of prime factors of a natural number . The number of distinct prime factors is assigned to () (little omega), while () (big omega) counts the total number of prime factors with multiplicity (see arithmetic function).
As an example, 60 2 = 3600 = 48 × 75, so as divisors of a power of 60 both 48 and 75 are regular. These numbers arise in several areas of mathematics and its applications, and have different names coming from their different areas of study. In number theory, these numbers are called 5-smooth, because they can be characterized as having only 2 ...