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Abundancy

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The abundancy of a positive integer
n
is defined as
abundancy (n) :=
σ (n)
n
 ,
where
σ (n)
is the sum of divisors of
n
.[1]

Since

σ (n)
n
 = 
d ∣n

d ∣n
d
n
 = 
d ∣n
d ∣n
  
d
n
 ,

an equivalent definition is

abundancy (n) := σ − 1(n)  = 
d ∣n
d ∣n
  
d   − 1.
A017665 Numerator of sum of reciprocals of divisors of
n
.
{1, 3, 4, 7, 6, 2, 8, 15, 13, 9, 12, 7, 14, 12, 8, 31, 18, 13, 20, 21, 32, 18, 24, 5, 31, 21, 40, 2, 30, 12, 32, 63, 16, 27, 48, 91, 38, 30, 56, 9, 42, 16, 44, 21, 26, 36, 48, 31, ...}
A017666 Denominator of sum of reciprocals of divisors of
n
.
{1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 6, 19, 10, 21, 11, 23, 2, 25, 13, 27, 1, 29, 5, 31, 32, 11, 17, 35, 36, 37, 19, 39, 4, 41, 7, 43, 11, 15, 23, 47, 12, 49, 50, ...}

Abundant numbers are numbers with abundancy greater than 2, perfect numbers are numbers with abundancy equal to 2 and deficient numbers are numbers with abundancy less than 2.

Multiply-perfect numbers

Main article page: Multiply-perfect numbers

When the abundancy of a positive integer
n
is a positive integer
kn, k   ≥   1,
we have a
k
-perfect number, 1 being the only 1-perfect number.
A007691 Multiply-perfect numbers:
n
divides
σ (n)
.
{1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, ...}
A054030
σ (n)
n
for
n
such that
σ (n)
is divisible by
n
.
{1, 2, 2, 3, 2, 3, 2, 4, 4, 3, 4, 4, 2, 4, 4, 3, 4, 3, 2, 5, 5, 4, 3, 4, 2, 4, 4, 5, 4, 5, 5, 4, 5, 5, 4, 4, 4, 5, 4, 4, 2, 5, 4, 5, 6, 5, 5, 5, 5, 5, 5, 6, 5, 5, 4, 5, 6, 5, 4, 4, 5, 4, 5, 4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 5, 6, 5, 6, 6, 5, 4, 4, 5, 4, 4, 5, 6, 5, 5, 4, 6, 4, 4, 6, 5, 6, 6, 6, 6, 6, 6, 6, 5, 6, ...}

See also


Notes

  1. Weisstein, Eric W., Abundancy, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Abundancy.html]