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superperfect number

In number theory, a superperfect number is a positive integer $n$ that satisfies
${\\displaystyle \\sigma ^{2}(n)=\\sigma (\\sigma (n))=2n\\,,}$
where $σ$ is the divisor summatory function. Superperfect numbers are not a generalization of perfect numbers, but have a common generalization. The term was coined by D. Suryanarayana (1969). The first few superperfect numbers are :
2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... (sequence A019279 in the OEIS).
To illustrate: it can be seen that 16 is a superperfect number as σ(16) = 1 + 2 + 4 + 8 + 16 = 31, and σ(31) = 1 + 31 = 32, thus σ(σ(16)) = 32 = 2 × 16. 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. 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. There are no odd superperfect numbers below 7×10²⁴.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Superperfect_number)

perfect number

nombre superparfait

francés

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http://data.loterre.fr/ark:/67375/PSR-FF87FBF1-H

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