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If a number is a squarefree positive integer, meaning that it is the product of some number of distinct prime numbers, then gives the number of different multiplicative partitions of . These are factorizations of N {\displaystyle N} into numbers greater than one, treating two factorizations as the same if they have the same factors in a ...
The Pisano period, denoted π (n), is the length of the period of this sequence. For example, the sequence of Fibonacci numbers modulo 3 begins: This sequence has period 8, so π (3) = 8. For n = 3, this is a visualization of the Pisano period in the two-dimensional state space of the recurrence relation.
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation. Given two positive numbers a and n , a modulo n (often abbreviated as a mod n ) is the remainder of the Euclidean division of a by n , where a is the dividend and n is the divisor .
Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed ...
The figure shows the 13 weak orderings on three elements. Starting from , the ordered Bell numbers are. 1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... (sequence A000670 in the OEIS). When the elements to be ordered are unlabeled (only the number of elements in each tied set matters, not their identities) what remains is a ...
The length of the repetend (period of the repeating decimal segment) of 1 / p is equal to the order of 10 modulo p. If 10 is a primitive root modulo p, then the repetend length is equal to p − 1; if not, then the repetend length is a factor of p − 1. This result can be deduced from Fermat's little theorem, which states that 10 p−1 ...
The sum of the subscripts in a monomial is equal to the total number of elements. Thus, the number of monomials that appear in the partial Bell polynomial is equal to the number of ways the integer n can be expressed as a summation of k positive integers. This is the same as the integer partition of n into k parts. For instance, in the above ...
Periodic table (electron configurations) Configurations of elements 109 and above are not available. Predictions from reliable sources have been used for these elements. Grayed out electron numbers indicate subshells filled to their maximum. Bracketed noble gas symbols on the left represent inner configurations that are the same in each period.