What is "abundant number"

Imperfect \Im*per"fect\, a. [L. imperfectus: pref. im- not + perfectus perfect: cf. F imparfait, whence OE. imparfit. See Perfect.]

1. Not perfect; not complete in all its parts; wanting a part; deective; deficient.

   Something he left imperfect in the state.
   --Shak.

   Why, then, your other senses grow imperfect.
   --Shak.

2. Wanting in some elementary organ that is essential to successful or normal activity.

   He . . . stammered like a child, or an amazed, imperfect person.
   --Jer. Taylor.

3. Not fulfilling its design; not realizing an ideal; not conformed to a standard or rule; not satisfying the taste or conscience; esthetically or morally defective.

   Nothing imperfect or deficient left Of all that he created.
   --Milton.

   Then say not man's imperfect, Heaven in fault; Say rather, man's as perfect as he ought.
   --Pope.

   Imperfect arch, an arch of less than a semicircle; a skew arch.

   Imperfect cadence (Mus.), one not ending with the tonic, but with the dominant or some other chord; one not giving complete rest; a half close.

   Imperfect consonances (Mus.), chords like the third and sixth, whose ratios are less simple than those of the fifth and forth.

   Imperfect flower (Bot.), a flower wanting either stamens or pistils.
   --Gray.

   Imperfect interval (Mus.), one a semitone less than perfect; as, an imperfect fifth.

   Imperfect number (Math.), a number either greater or less than the sum of its several divisors; in the former case, it is called also a defective number; in the latter, an abundant number.

   Imperfect obligations (Law), obligations as of charity or gratitude, which cannot be enforced by law.

   Imperfect power (Math.), a number which can not be produced by taking any whole number or vulgar fraction, as a factor, the number of times indicated by the power; thus, 9 is a perfect square, but an imperfect cube.

   Imperfect tense (Gram.), a tense expressing past time and incomplete action.

n. (context number theory English) A number that is less than the sum of all of its divisors except itself.

In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example.