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A perfect power has a common divisor m > 1 for all multiplicities (it is of the form a m for some a > 1 and m > 1). The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 (sequence A001597 in the OEIS). 1 is sometimes included. A powerful number (also called squareful) has multiplicity above 1 for all prime
16, 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, ... The length of an interval of consecutive integers with property that every element has a factor in common with one of the endpoints. A059756
Not all e i ≡ 2 (mod 5). [42] If all e i ≡ 1 (mod 3) or 2 (mod 5), then the smallest prime factor of N must lie between 10 8 and 10 1000. [42] More generally, if all 2e i +1 have a prime factor in a given finite set S, then the smallest prime factor of N must be smaller than an effectively computable constant depending only on S. [42]
The number of domino tilings of a 4×4 checkerboard is 36. [10] Since it is possible to find sequences of 36 consecutive integers such that each inner member shares a factor with either the first or the last member, 36 is an Erdős–Woods number. [11] The sum of the integers from 1 to 36 is 666 (see number of the beast). 36 is also a ...
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. [3] [4] [5] For example,
Elementary arithmetic is a branch of mathematics involving addition, subtraction, multiplication, and division. Due to its low level of abstraction, broad range of application, and position as the foundation of all mathematics, elementary arithmetic is generally the first branch of mathematics taught in schools. [1] [2]