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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.
120 6: 720 7: 5 040: 8: 40 320: 9: 362 880: 10: 3 628 ... 70 1.197 857 167 × 10 100 ... matching to within a constant factor the time for fast multiplication ...
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 ...
For n ≥ 2, a(n) is the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached; a(n) = −1 if no prime is ever reached. A037274
Order HCN n prime factorization prime exponents number of prime factors d(n) primorial factorization 1 1: 0 1 2 2* : 1 1 2 3 4: 2 2 3 4 6* : 1,1 2 4 5 12* : 2,1 3 6 6 24
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, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number.
120, 672, 523776, 459818240, 1476304896, 51001180160 (sequence A005820 in the OEIS) If there exists an odd perfect number m (a famous open problem) then 2m would be 3-perfect, since σ(2m) = σ(2) σ(m) = 3×2m. An odd triperfect number must be a square number exceeding 10 70 and have at least 12 distinct prime factors, the largest exceeding 10 ...