info prev up next book cdrom email home

Hyperperfect Number

A number $n$ is called $k$-hyperperfect if

n&=&1+k\sum_i d_i\\

where the summation is over the Proper Divisors with $1<d_i<n$, giving


where $\sigma(n)$ 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.


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.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

© 1996-9 Eric W. Weisstein