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As a biprime with proper divisors 1, 3 and 7, twenty-one has a prime aliquot sum of 11 within an aliquot sequence containing only one composite number (21, 11, 1, 0); it is the second composite number with an aliquot sum of 11, following 18. 21 is the first member of the second cluster of consecutive discrete semiprimes (21, 22), where the next such cluster is (33, 34, 35).
For instance, 1/3+1/4 = 7/12, so a notation like would represent the number that would now more commonly be written as the mixed number , or simply the improper fraction . Notation of this form can be distinguished from sequences of numerators and denominators sharing a fraction bar by the visible break in the bar.
The number in the n-th month is the n-th Fibonacci number. [21] The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas. [22] Solution to Fibonacci rabbit problem: In a growing idealized population, the number of rabbit pairs form the Fibonacci sequence.
A repfigit, or Keith number, is an integer such that, when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 (4, 7, 11, 18, 29, 47) reaches 47.
For instance, the Zeckendorf representation of 19 is 101001 (where the 1's mark the positions of the Fibonacci numbers used in the expansion 19 = 13 + 5 + 1), the binary sequence 101001, interpreted as a binary number, represents 41 = 32 + 8 + 1, and the 19th fibbinary number is 41.
64 = 34 + 21 + 5 + 3 + 1 64 = 34 + 13 + 8 + 5 + 3 + 1. but these are not Zeckendorf representations because 34 and 21 are consecutive Fibonacci numbers, as are 5 and 3. For any given positive integer, its Zeckendorf representation can be found by using a greedy algorithm, choosing the largest possible Fibonacci number at each stage.
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Fibonacci coding is a closely related numeration system used for integers. In this system, only digits 0 and 1 are used and the place values of the digits are the Fibonacci numbers. As with base-φ, the digit sequence "11" is avoided by rearranging to a standard form, using the Fibonacci recurrence relation F k+1 = F k + F k−1. For example,