Number whose divisors summed twice over equal twice itself
In number theory, a superperfect number is a positive integer n that satisfies
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).[1]
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, 2k, such that 2k+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] There are no odd superperfect numbers below 7×1024.[1]
^ abcGuy (2004) p. 99.
^ abWeisstein, Eric W. "Superperfect Number". MathWorld.
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