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d() is the number of positive divisors of n, including 1 and n itself; σ() is the sum of the positive divisors of n, including 1 and n itselfs() is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n
Many properties of a natural number n can be seen or directly computed from the prime factorization of n. The multiplicity of a prime factor p of n is the largest exponent m for which p m divides n. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since p = p 1).
It can also be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2, 3, −3.
2.42 Perrin primes. 2.43 ... write the prime factorization of n in base 10 and concatenate the factors; iterate until a prime is ... 20 p − 1 ≡ 1 (mod p 2 ...
For example, if n is 24, there are two prime factors (p 1 is 2; p 2 is 3); noting that 24 is the product of 2 3 ×3 1, a 1 is 3 and a 2 is 1. Thus we can calculate σ 0 ( 24 ) {\displaystyle \sigma _{0}(24)} as so:
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:
The smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5, 7, 11, 13, 17, 19, 23, and 29 (sequence A047802 in the OEIS). An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. [1]
The factorizations are often not unique in the sense that the unit could be absorbed into any other factor with exponent equal to one. ... 11+20i 20+11i (p) (p) 522: ...