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Primary pseudoperfect number
In mathematics, and particularly in number theory, N is a primary pseudoperfect number if it satisfies the Egyptian fraction equation
where the sum is over only the prime divisors of N.
Equivalently, N is a primary pseudoperfect number if it satisfies
Except for the primary pseudoperfect number N = 2, this expression gives a representation for N as the sum of distinct divisors of N. Therefore, each primary pseudoperfect number N (except N = 2) is also pseudoperfect.
The eight known primary pseudoperfect numbers are
The first four of these numbers are one less than the corresponding numbers in Sylvester's sequence, but then the two sequences diverge.
It is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are any odd primary pseudoperfect numbers.
The prime factors of primary pseudoperfect numbers sometimes may provide solutions to Znám's problem, in which all elements of the solution set are prime. For instance, the prime factors of the primary pseudoperfect number 47058 form the solution set {2,3,11,23,31} to Znám's problem. However, the smaller primary pseudoperfect numbers 2, 6, 42, and 1806 do not correspond to solutions to Znám's problem in this way, as their sets of prime factors violate the requirement that no number in the set can equal one plus the product of the other numbers. Anne (1998) observes that there is exactly one solution set of this type that has k primes in it, for each k ≤ 8, and conjectures that the same is true for larger k.
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Primary pseudoperfect number
In mathematics, and particularly in number theory, N is a primary pseudoperfect number if it satisfies the Egyptian fraction equation
where the sum is over only the prime divisors of N.
Equivalently, N is a primary pseudoperfect number if it satisfies
Except for the primary pseudoperfect number N = 2, this expression gives a representation for N as the sum of distinct divisors of N. Therefore, each primary pseudoperfect number N (except N = 2) is also pseudoperfect.
The eight known primary pseudoperfect numbers are
The first four of these numbers are one less than the corresponding numbers in Sylvester's sequence, but then the two sequences diverge.
It is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are any odd primary pseudoperfect numbers.
The prime factors of primary pseudoperfect numbers sometimes may provide solutions to Znám's problem, in which all elements of the solution set are prime. For instance, the prime factors of the primary pseudoperfect number 47058 form the solution set {2,3,11,23,31} to Znám's problem. However, the smaller primary pseudoperfect numbers 2, 6, 42, and 1806 do not correspond to solutions to Znám's problem in this way, as their sets of prime factors violate the requirement that no number in the set can equal one plus the product of the other numbers. Anne (1998) observes that there is exactly one solution set of this type that has k primes in it, for each k ≤ 8, and conjectures that the same is true for larger k.