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The sum of its factors (including one and itself) sum to 360, exactly three times 120. Perfect numbers are order two ( 2-perfect ) by the same definition. 120 is the sum of a twin prime pair (59 + 61) and the sum of four consecutive prime numbers (23 + 29 + 31 + 37), four consecutive powers of two (8 + 16 + 32 + 64), and four consecutive powers ...
The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). 0! = 1 is sometimes included. A k-smooth number (for a natural number k) has its prime factors ≤ k (so it is also j-smooth for any j > k). m is smoother than n if the largest prime factor of m is below the largest of n.
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial ...
For example, if n = 171 × p × q where p < q are very large primes, trial division will quickly produce the factors 3 and 19 but will take p divisions to find the next factor. As a contrasting example, if n is the product of the primes 13729, 1372933, and 18848997161, where 13729 × 1372933 = 18848997157, Fermat's factorization method will ...
If none of its prime factors are repeated, it is called squarefree. (All prime numbers and 1 are squarefree.) For example, 72 = 2 3 × 3 2, all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree. Euler diagram of numbers under 100:
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: 7 = 7 1, 9 = 3 2 and 64 = 2 6 are prime powers, while 6 = 2 × 3, 12 = 2 2 × 3 and 36 = 6 2 = 2 2 × 3 2 are not. The sequence of prime powers begins:
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The most widely used of these [67] uses the gamma function, which can be defined for positive real numbers as the integral =. The resulting function is related to the factorial of a non-negative integer n {\displaystyle n} by the equation n ! = Γ ( n + 1 ) , {\displaystyle n!=\Gamma (n+1),} which can be used as a definition of the factorial ...