Deficient Number

Numbers which are not Perfect and for which

$\begin{displaymath} s(N)\equiv\sigma(N)-N < N, \end{displaymath}$

or equivalently

$\begin{displaymath} \sigma(n)<2n, \end{displaymath}$

where $\sigma(N)$ is the Divisor Function. Deficient numbers are sometimes called Defective Numbers (Singh 1997). Primes, Powers of Primes, and any divisors of a Perfect or deficient number are all deficient. The first few deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, ... (Sloane's A005100).

References

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 3-33, 1952.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 45, 1994.

Singh, S. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. New York: Walker, p. 11, 1997.

Sloane, N. J. A. Sequence A005100/M0514 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.