Semiperfect Number

A number such as $20=1+4+5+10$ which is the Sum of some (or all) of its Proper Divisors. A semiperfect number which is the Sum of all its Proper Divisors is called a Perfect Number. The first few semiperfect numbers are 6, 12, 18, 20, 24, 28, 30, 36, 40, ... (Sloane's A005835). Every multiple of a semiperfect number is semiperfect, as are all numbers $2^mp$ for $m\geq 1$ and $p$ a Prime between $2^m$ and $2^{m+1}$ (Guy 1994, p. 47).

A semiperfect number cannot be Deficient. Rare Abundant Numbers which are not semiperfect are called Weird Numbers. Semiperfect numbers are sometimes also called Pseudoperfect Numbers.

See also Abundant Number, Deficient Number, Perfect Number, Primitive Semiperfect Number, Weird Number

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. Sequence A005835/M4094 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.

Zachariou, A. and Zachariou, E. ``Perfect, Semi-Perfect and Ore Numbers.'' Bull. Soc. Math. Gréce (New Ser.) 13, 12-22, 1972.