A number is called -hyperperfect if

where the summation is over the Proper Divisors with , giving

where is the Divisor Function. The first few hyperperfect numbers are 21, 301, 325, 697, 1333, ... (Sloane's A007592). 2-hyperperfect numbers include 21, 2133, 19521, 176661, ... (Sloane's A007593), and the first 3-hyperperfect number is 325.

**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.

Sloane, N. J. A. Sequences
A007592/M5113
and A007593/M5121
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-25