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Superperfect Number

A superperfect number is a positive integer n such that

sigma^2(n)=sigma(sigma(n))=2n,

where sigma(n) is the divisor function giving the sum of the positive divisors of n. Even superperfect numbers are just 2^(p-1), where M_p=2^p-1 is a Mersenne prime. If any odd superperfect numbers exist, they are square numbers and either n or sigma(n) is divisible by at least three distinct primes (Dandapat et al. 1975, pp. 360 and 363).

For positive integers m and k, write sigma^0(n)=n and sigma^m(n)=sigma(sigma^(m-1)(n)). More generally, an m-superperfect (or (m, 2)-superperfect) number is a positive integer n for which sigma^m(n)=2n, and an (m,k)-perfect number is a positive integer n for which sigma^m(n)=kn. A number n can be tested to see if it is (m,k)-perfect using the following Wolfram Language code: 

  SuperperfectQ[m_, n_, k_:2] :=
    Nest[DivisorSigma[1, #]&, n, m] == k n

The first few (2, 2)-perfect numbers are 2, 4, 16, 64, 4096, 65536, 262144, ... (OEIS A019279; Cohen and te Riele 1996, p. 93). For m>=3, there are no even m-superperfect numbers (Guy 1994, p. 65). On the basis of computer searches, J. McCranie reported that there are no (m,2)-superperfect numbers less than 4.29×10^9 for any m>=3 (J. McCranie, pers. comm., Nov. 11, 2001). McCranie further conjectured that there are no (m,2)-superperfect numbers for m>3.

See also

Divisor Function, Mersenne Number, Perfect Number

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References

Cohen, G. L. and te Riele, H. J. J. "Iterating the Sum-of-Divisors Function." Experim. Math. 5, 91-100, 1996. https://doi.org/10.1080/10586458.1996.10504580.Dandapat, G. G.; Hunsucker, J. L.; and Pomerance, C. "Some New Results on Odd Perfect Numbers." Pacific J. Math. 57, 359-364, 1975. https://msp.org/pjm/1975/57-2/pjm-v57-n2-p07-p.pdf.Guy, R. K. "Superperfect Numbers." §B9 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 65-66, 1994.Kanold, H.-J. "Über 'Super Perfect Numbers.' " Elem. Math. 24, 61-62, 1969.Lord, G. "Even Perfect and Superperfect Numbers." Elem. Math. 30, 87-88, 1975.Sloane, N. J. A. Sequence A019279 in "The On-Line Encyclopedia of Integer Sequences."Suryanarayana, D. "Super Perfect Numbers." Elem. Math. 24, 16-17, 1969.Suryanarayana, D. "There Is No Odd Super Perfect Number of the Form p^(2alpha)." Elem. Math. 28, 148-150, 1973.

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Superperfect Number

Cite this as:

Weisstein, Eric W. "Superperfect Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SuperperfectNumber.html

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