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Hyperperfect Number


A number n is called k-hyperperfect if

n=1+ksum_(i)d_i
(1)
=1+k[sigma(n)-n-1],
(2)

where sigma(n) is the divisor function and the summation is over the proper divisors with 1<d_i<n. Rearranging gives

 ksigma(n)=(k+1)n+k-1.
(3)

Taking k=1 gives the usual perfect numbers.

If k>1 is an odd integer, and p=(3k+1)/2 and q=3k+4=2p+3 are prime, then p^2q is k-hyperperfect. McCranie (2000) conjectures that all k-hyperperfect numbers for odd k>1 are in fact of this form. Similarly, if p and q are distinct odd primes such that k(p+q)=pq-1 for some integer k, then n=pq is k-hyperperfect. Finally, if k>0 and p=k+1 is prime, then if q=p^i-p+1 is prime for some i>1< then n=p^(i-1)q is k-hyperperfect (McCranie 2000).

The first few hyperperfect numbers (excluding perfect numbers) are 21, 301, 325, 697, 1333, ... (OEIS A007592). If perfect numbers are included, the first few are 6, 21, 28, 301, 325, 496, ... (OEIS A034897), whose corresponding values of k are 1, 2, 1, 6, 3, 1, 12, ... (OEIS A034898). The following table gives the first few k-hyperperfect numbers for small values of k. McCranie (2000) has tabulated all hyperperfect numbers less than 10^(11).

kOEISk-hyperperfect number
1A0003966 ,28, 496, 8128, ...
2A00759321, 2133, 19521, 176661, ...
3325, ...
41950625, 1220640625, ...
6A028499301, 16513, 60110701, ...
10159841, ...
1110693, ...
12A028500697, 2041, 1570153, 62722153, ...

See also

Perfect Number

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References

Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." §B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-53, 1994.McCranie, J. S. "A Study of Hyperperfect Numbers." J. Integer Sequences 3, No. 00.1.3, 2000. http://www.math.uwaterloo.ca/JIS/VOL3/VOL3/mccranie.Minoli, D. "Issues in Nonlinear Hyperperfect Numbers." Math. Comput. 34, 639-645, 1980.Roberts, J. The Lure of the Integers. Washington, DC: Math. Assoc. Amer., p. 177, 1992.Sloane, N. J. A. Sequences A000396/M4186, A007592/M5113, A007593/M5121, A028499, A028500, A034897, and A034898 in "The On-Line Encyclopedia of Integer Sequences."te Riele, H. J. J. "Hyperperfect Numbers with Three Different Prime Factors." Math. Comput. 36, 297-298, 1981.

Referenced on Wolfram|Alpha

Hyperperfect Number

Cite this as:

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

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