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Keith numbers are rare and hard to find. They can be found by exhaustive search, and no more efficient algorithm is known. [ 2 ] According to Keith, in base 10 , on average 9 10 log 2 10 ≈ 2.99 {\displaystyle \textstyle {\frac {9}{10}}\log _{2}{10}\approx 2.99} Keith numbers are expected between successive powers of 10 . [ 3 ]
The smallest integer m > 1 such that p n # + m is a prime number, where the primorial p n # is the product of the first n prime numbers. A005235 Semiperfect numbers
A prime number is a natural number that has no natural number divisors other than the number 1 and itself.. To find all the prime numbers less than or equal to a given integer N, a sieve algorithm examines a set of candidates in the range 2, 3, …, N, and eliminates those that are not prime, leaving the primes at the end.
For example, John von Neumann constructs the number 0 as the empty set {}, and the successor of n, S(n), as the set n ∪ {n}. The axiom of infinity then guarantees the existence of a set that contains 0 and is closed with respect to S. The smallest such set is denoted by N, and its members are called natural numbers. [2]
This example is mutual single recursion, and could easily be replaced by iteration. In this example, the mutually recursive calls are tail calls, and tail call optimization would be necessary to execute in constant stack space. In C, this would take O(n) stack space, unless rewritten to use jumps instead of calls. [4]
The natural integer 6174 is known as Kaprekar's constant, [1] [2] [3] after the Indian mathematician D. R. Kaprekar. This number is notable for the following curious behavior: Select any four-digit number which has at least two different digits (leading zeros are allowed), Create two new four-digit numbers by arranging the original digits in a.
A real number is computable if and only if the set of natural numbers it represents (when written in binary and viewed as a characteristic function) is computable. The set of computable real numbers (as well as every countable, densely ordered subset of computable reals without ends) is order-isomorphic to the set of rational numbers.
a n a n−1...a 1 a 0.c 1 c 2 c 3... is the original representation in the original numeral system. b is the original radix. b is 10 if converting from decimal. a k and c k are the digits k places to the left and right of the radix point respectively. For instance,