In mathematics, a Keith number or repfigit number is an integer that appears as a term in a linear recurrence relation with initial terms based off its own digits. Given an n-digit number
a sequence SN is formed with initial terms and with a general term produced as the sum of the previous n terms. If the number N appears in the sequence SN, then N is said to be a Keith number.
For example, taking 197 in such a way creates the sequence . The first few Keith numbers are
Whether or not there are infinitely many Keith numbers is currently a matter of speculation. There are only 71 Keith numbers below 1019, making them much rarer than prime numbers.
External link
Keith numbers (sequence A007629) (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A007629) in OEIS
I introduced them in a paper in 1987 (where they were called repfigit numbers or repfigits) and they proved popular enough to inspire several papers by other authors, most notably a whole series of papers that appeared in 1994 in Volume 26, Number 3 of the Journal of Recreational Mathematics.
These numbers are in some ways reminscent of the primes in their unpredictable appearance among the integers.
Define a cluster of Keithnumbers as a set of two or more (all with the same number of digits) in which all the numbers are integer multiples of the smallest one in the set.
The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.
In music Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or formal elements.
The Fibonacci numbers are also apparent in the organisation of the sections in the music of Debussy's Image, Reflections in Water, in which the sequence of keys is marked out by the intervals 34, 21, 13 and 8.
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