Hyperperfect Number
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. SequencesA007592/M5113and A007593/M5121in ``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.
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