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In number theory, a weird number is a natural number that is abundant but not semiperfect. [ 1 ] [ 2 ] In other words, the sum of the proper divisors ( divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.
The odious numbers give the positions of the nonzero values in the Thue–Morse sequence. Every power of two is odious, because its binary expansion has only one nonzero bit. Except for 3, every Mersenne prime is odious, because its binary expansion consists of an odd prime number of consecutive nonzero bits.
In number theory, an evil number is a non-negative integer that has an even number of 1s in its binary expansion. [1] These numbers give the positions of the zero values in the Thue–Morse sequence, and for this reason they have also been called the Thue–Morse set. [2] Non-negative integers that are not evil are called odious numbers.
An abundant number which is not a semiperfect number is called a weird number. [6] An abundant number with abundance 1 is called a quasiperfect number, although none have yet been found. Every abundant number is a multiple of either a perfect number or a primitive abundant number.
Bubble sort, sometimes referred to as sinking sort, is a simple sorting algorithm that repeatedly steps through the input list element by element, comparing the current element with the one after it, swapping their values if needed.
The odd–even sort algorithm correctly sorts this data in passes. (A pass here is defined to be a full sequence of odd–even, or even–odd comparisons. The passes occur in order pass 1: odd–even, pass 2: even–odd, etc.) Proof: This proof is based loosely on one by Thomas Worsch. [6]
The following version is by Philippe Flajolet and Robert Sedgewick: [1] The director of a prison offers 100 death row prisoners, who are numbered from 1 to 100, a last chance. A room contains a cupboard with 100 drawers. The director randomly puts one prisoner's number in each closed drawer. The prisoners enter the room, one after another.
In number theory, a deficient number or defective number is a positive integer n for which the sum of divisors of n is less than 2n. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than n. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient.