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The numbers of compositions of n +1 into k +1 ordered partitions form Pascal's triangle Using the Fibonacci sequence to count the {1, 2}-restricted compositions of n, for example, the number of ways one can ascend a staircase of length n, taking one or two steps at a time
Demonstration, with Cuisenaire rods, of the first four highly composite numbers: 1, 2, 4, 6. A highly composite number is a positive integer that has more divisors than all smaller positive integers. If d(n) denotes the number of divisors of a positive integer n, then a positive integer N is highly composite if d(N) > d(n) for all n < N.
To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers 2, 3, 5, and so on, up to the square root of n. For larger numbers, especially when using a computer, various more sophisticated factorization algorithms are more efficient.
By taking conjugates, the number p k (n) of partitions of n into exactly k parts is equal to the number of partitions of n in which the largest part has size k. The function p k (n) satisfies the recurrence p k (n) = p k (n − k) + p k−1 (n − 1) with initial values p 0 (0) = 1 and p k (n) = 0 if n ≤ 0 or k ≤ 0 and n and k are not both ...
Algorithms are known for decomposing univariate polynomials in polynomial time. Polynomials which are decomposable in this way are composite polynomials; those which are not are indecomposable polynomials or sometimes prime polynomials [1] (not to be confused with irreducible polynomials, which cannot be factored into products of polynomials).
The number of representable numbers in the range from 0 to any number is proportional to , with a limiting constant of proportionality given by the Landau–Ramanujan constant, approximately 0.764. [3] The product of any two representable numbers is another representable number.
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A number n that has more divisors than any x < n is a highly composite number (though the first two such numbers are 1 and 2). Composite numbers have also been called "rectangular numbers", but that name can also refer to the pronic numbers, numbers that are the product of two consecutive integers.