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Sociable Dudeney numbers and amicable Dudeney numbers are the powers of their respective roots. The number of iterations i {\displaystyle i} needed for F p , b i ( n ) {\displaystyle F_{p,b}^{i}(n)} to reach a fixed point is the Dudeney function's persistence of n {\displaystyle n} , and undefined if it never reaches a fixed point.
A trivial example of precomputation is the use of hardcoded mathematical constants, such as π and e, rather than computing their approximations to the necessary precision at run time. In databases , the term materialization is used to refer to storing the results of a precomputation, [ 1 ] [ 2 ] such as in a materialized view .
Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is
The cube operation can also be defined for any other mathematical expression, for example (x + 1) 3. The cube is also the number multiplied by its square: n 3 = n × n 2 = n × n × n. The cube function is the function x ↦ x 3 (often denoted y = x 3) that maps a number to its cube. It is an odd function, as (−n) 3 = −(n 3). The volume of ...
Figurate numbers were a concern of the Pythagorean worldview. It was well understood that some numbers could have many figurations, e.g. 36 is a both a square and a triangle and also various rectangles. The modern study of figurate numbers goes back to Pierre de Fermat, specifically the Fermat polygonal number theorem.
A more restrictive taxicab problem requires that the taxicab number be cubefree, which means that it is not divisible by any cube other than 1 3. When a cubefree taxicab number T is written as T = x 3 + y 3, the numbers x and y must be relatively prime. Among the taxicab numbers Ta(n) listed above, only Ta(1) and Ta(2) are cubefree
i is the number of initial rounds; r is the number of rounds per block; b is the block size in bytes, defined for {1, 2, 3, ... 128} f is the number of final rounds; h is the size of the hash output in bits, defined for {8, 16, 24, 32, ... 512} In the original NIST submission, i and f was fixed to 10r.
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the triangle inequality as well: [2] Indeed, if we fix three words a, b and c, then whenever there is a ...