abundant number
التعريفات والمعاني
== English ==
=== Pronunciation ===
(Received Pronunciation) IPA(key): /əˈbʌn.dn̩t ˈnʌm.bə/
(US) IPA(key): /əˈbʌn.dn̩t ˈnʌm.bɚ/, /əˈbn̩.dn̩t ˈnʌm.bɚ/
=== Noun ===
abundant number (plural abundant numbers)
(number theory) A number that is less than the sum of its proper divisors (all divisors except the number itself).
Synonym: excessive number
1992, Stanley Rabinowitz (editor), Index to Mathematical Problems, 1980-1984, MathPro Press, page 185,
(a) Let k be fixed. Do there exist sequences of k consecutive abundant numbers?
==== Usage notes ====
The requirement may be expressed as
s
(
n
)
>
n
{\displaystyle s(n)>n}
, where
s
(
n
)
{\displaystyle s(n)}
denotes the aliquot sum (sum of proper divisors) of
n
{\displaystyle n}
.
It is also sometimes expressed as
σ
(
n
)
>
2
n
{\displaystyle \sigma (n)>2n}
, where
σ
(
n
)
{\displaystyle \sigma (n)}
(sometimes
σ
1
(
n
)
{\displaystyle \sigma _{1}(n)}
) denotes the sum of all divisors of
n
{\displaystyle n}
.
Given an abundant number
n
{\displaystyle n}
, the amount,
s
(
n
)
−
n
,
{\displaystyle s(n)-n,}
by which the aliquot sum exceeds it may be called its abundance.
For arbitrary
n
{\displaystyle n}
, the ratio
σ
(
n
)
n
{\displaystyle {\frac {\sigma (n)}{n}}}
may be called its abundancy index. Thus, an abundant number is one whose abundancy index is > 2.
==== Hyponyms ====
weird number
==== Derived terms ====
==== Translations ====
=== See also ===
abundance
abundancy index
amicable number
deficient number
perfect number
semiperfect number
sociable number
=== Further reading ===
Divisor function on Wikipedia.Wikipedia
abundant number on The Prime Glossary
Abundant Number on Wolfram MathWorld