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A divisor d of n is a bi-unitary divisor if the greatest common unitary divisor of d and n/d is 1. This concept originates from D. Suryanarayana (1972). [The number of bi-unitary divisors of an integer, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics 251: 273–282, New York, Springer–Verlag].
A divisor d of a positive integer n is called a bi-unitary divisor of n if the greatest common unitary divisor (gcud) of d and n/d equals 1. This concept is due to D. Surynarayana (1972). The sum of the (positive) bi-unitary divisors of n is denoted by σ ** (n). Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers ...
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. ... 1, 7, 11, 77 4 ...
The number 60 is a unitary perfect number because 1, 3, 4, 5, 12, 15, and 20 are its proper unitary divisors, and 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60. The first five, and only known, unitary perfect numbers are:
The number 1 is called a unit. ... = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23 ... The number of divisors can be computed by increasing all multiplicities by 1 and ...
The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle. If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6.
For example, to find the Hall divisors of 60, its prime power factorization is 2 2 × 3 × 5, so one takes any product of 3, 2 2 = 4, and 5. Thus, the Hall divisors of 60 are 1, 3, 4, 5, 12, 15, 20, and 60. A Hall subgroup of G is a subgroup whose order is a Hall divisor of the order of G. In other words, it is a subgroup whose order is coprime ...
If n is an even superperfect number, then n must be a power of 2, 2 k, such that 2 k+1 − 1 is a Mersenne prime. [1] [2] It is not known whether there are any odd superperfect numbers. An odd superperfect number n would have to be a square number such that either n or σ(n) is divisible by at least three distinct primes. [2]