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The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after Eugène Catalan, though they were previously discovered in the 1730s by Minggatu. The n-th Catalan number can be expressed directly in terms of the central binomial coefficients by
The sequence begins: 1, 1, 2, 5, 14, 42, 132, ... The Catalan numbers are solutions to numerous counting problems which often have a recursive flavour. In fact, one author lists over 60 different possible interpretations of these numbers. For example, the n th Catalan number is the number of full binary trees with n internal nodes, or n+1 leaves.
A number that has the same number of digits as the number of digits in its prime factorization, including exponents but excluding exponents equal to 1. A046758: Extravagant numbers: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, ... A number that has fewer digits than the number of digits in its prime factorization (including ...
A general problem is to count the number of balanced brackets (or legal permutations) that a string of m open and m closed brackets forms (total of 2m brackets). By legally arranged, the following rules apply: For the sequence as a whole, the number of open brackets must equal the number of closed brackets
The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ 1 or the aliquot sum function s in the following way: [1] = = = > = = = If the s n-1 = 0 condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are convergent, the limit of these ...
The sequence of Catalan numbers and the sequence of Schröder–Hipparchus numbers, viewed as infinite-dimensional vectors, are the unique eigenvectors for the first two in a sequence of naturally defined linear operators on number sequences.
Angel numbers are repeating number sequences, often used as guides for deeper spiritual exploration. Ranging from 000 to 999, each sequence carries its own distinct meaning and energy.
Catalan discovered this sequence after the discovery of the primality of = by Lucas in 1876. [ 1 ] [ 5 ] Catalan conjectured that they are prime "up to a certain limit". Although the first five terms are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge.