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Concept
Nombre parfait multiple
Résumé
In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.
For a given natural number k, a number n is called k-perfect (or k-fold perfect) if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number. As of 2014, k-perfect numbers are known for each value of k up to 11.
It is unknown whether there are any odd multiply perfect numbers other than 1. The first few multiply perfect numbers are:
1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, ... .
The sum of the divisors of 120 is
1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 + 120 = 360
which is 3 × 120. Therefore 120 is a 3-perfect number.
The following table gives an overview of the smallest known k-perfect numbers for k ≤ 11 :
It can be proven that:
For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p + 1)-perfect. This implies that an integer n is a 3-perfect number divisible by 2 but not by 4, if and only if n/2 is an odd perfect number, of which none are known.
If 3n is 4k-perfect and 3 does not divide n, then n is 3k-perfect.
It is unknown whether there are any odd multiply perfect numbers other than 1. However if an odd k-perfect number n exists where k > 2, then it must satisfy the following conditions:
The largest prime factor is ≥ 100129
The second largest prime factor is ≥ 1009
The third largest prime factor is ≥ 101
In little-o notation, the number of multiply perfect numbers less than x is for all ε > 0.
The number of k-perfect numbers n for n ≤ x is less than , where c and c are constants independent of k.
Under the assumption of the Riemann hypothesis, the following inequality is true for all k-perfect numbers n, where k > 3
where is Euler's gamma constant. This can be proven using Robin's theorem.
Source officielle
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