Editing Abundant number (section)

==Related concepts==
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Numbers whose sum of proper factors equals the number itself (such as 6 and 28) are called [[perfect number]]s, while numbers whose sum of proper factors is less than the number itself are called [[deficient number]]s. The first known classification of numbers as deficient, perfect or abundant was by [[Nicomachus]] in his ''[[Introduction to Arithmetic|Introductio Arithmetica]]'' (circa 100 AD), which described abundant numbers as like deformed animals with too many limbs.

The '''abundancy index''' of ''n'' is the ratio ''σ''(''n'')/''n''.<ref>{{cite journal | last=Laatsch | first=Richard | title=Measuring the abundancy of integers | journal=[[Mathematics Magazine]] | volume=59 | number=2 | pages=84–92 | year=1986 | issn=0025-570X | zbl=0601.10003 |jstor=2690424 |mr=0835144 | doi=10.2307/2690424}}</ref> Distinct numbers ''n''<sub>1</sub>, ''n''<sub>2</sub>, ... (whether abundant or not) with the same abundancy index are called [[friendly number]]s.

The sequence (''a''<sub>''k''</sub>) of least numbers ''n'' such that ''σ''(''n'') > ''kn'', in which ''a''<sub>2</sub> = 12 corresponds to the first abundant number, grows very quickly {{OEIS|id=A134716}}.

The smallest odd integer with abundancy index exceeding 3 is 1018976683725 = 3<sup>3</sup> × 5<sup>2</sup> × 7<sup>2</sup> × 11 × 13 × 17 × 19 × 23 × 29.<ref>For smallest odd integer ''k'' with abundancy index exceeding ''n'', see {{Cite OEIS|sequencenumber=A119240|name=Least odd number ''k'' such that sigma(k)/k >= n.}}</ref>

If '''p''' = (''p''<sub>1</sub>, ..., ''p<sub>n</sub>'') is a list of primes, then '''p''' is termed ''abundant'' if some integer composed only of primes in '''p''' is abundant.  A necessary and sufficient condition for this is that the product of ''p<sub>i</sub>''/(''p<sub>i</sub>'' − 1) be > 2.<ref>{{cite journal | title = Sums of divisors and Egyptian fractions | last = Friedman | first = Charles N. | journal = [[Journal of Number Theory]] | year = 1993 | volume = 44 | pages = 328–339 | mr = 1233293 | zbl = 0781.11015 | doi = 10.1006/jnth.1993.1057 | issue = 3 | doi-access = free }}</ref>