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Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are in bold and superior highly composite numbers are starred. ... 5, 7, 10, 11 ...
the k given prime numbers p i must be precisely the first k prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than n with the same number of divisors (for instance 10 = 2 × 5 may be replaced with 6 = 2 × 3; both have four divisors);
Divisor function d(n) up to n = 250 Prime-power factors In number theory , a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors . Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power.
Divisor function σ 0 (n) up to n = 250 Sigma function σ 1 (n) up to n = 250 Sum of the squares of divisors, σ 2 (n), up to n = 250 Sum of cubes of divisors, σ 3 (n) up to n = 250. In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer.
gcd(m, n) (greatest common divisor of m and n) is the product of all prime factors which are both in m and n (with the smallest multiplicity for m and n). m and n are coprime (also called relatively prime) if gcd( m , n ) = 1 (meaning they have no common prime factor).
The elements 2 and 1 + √ −3 are two maximal common divisors (that is, any common divisor which is a multiple of 2 is associated to 2, the same holds for 1 + √ −3, but they are not associated, so there is no greatest common divisor of a and b.
This function measures the tendency of divisors of a number to cluster. The growth of this sequence is limited by Δ ( m n ) ≤ Δ ( n ) d ( m ) {\displaystyle \Delta (mn)\leq \Delta (n)d(m)} where d ( n ) {\displaystyle d(n)} is the number of divisors of n {\displaystyle n} .
To compute the largest power of 2 dividing the binomial coefficient () write m = 3 and n − m = 7 in base p = 2 as 3 = 11 2 and 7 = 111 2.Carrying out the addition 11 2 + 111 2 = 1010 2 in base 2 requires three carries: