Abundant numbers

The abundant numbers are positive integers n for which the sum of divisors of n exceeds  

2n

.
The first even abundant number is  

12 = 2 2  ⋅   3

, with  

σ (12) = 2 3  −  1 2  −  1  ⋅   (3 + 1) = 7  ⋅   4 = 28 > 24 = 2  ⋅   12

.
The first odd abundant number (the 232^nd abundant number) is  

945 = 3 3  ⋅   5  ⋅   7 = 1  ⋅   3  ⋅   5  ⋅   7  ⋅   9 = 9!!

(the double factorial of 9), with  

σ (945) = 3 4  −  1 3  −  1  ⋅   (5 + 1)  ⋅   (7 + 1) = 40  ⋅   6  ⋅   8 = 1920 > 1890 = 2  ⋅   945

.

The abundancy of n is

$\mathrm{abundancy}(n):=\frac{\sigma (n)}{n},$

where  

σ (n)

is the sum of divisors of  

n

. An equivalent definition is

$\mathrm{abundancy}(n)={\sigma }_{-1}(n):=\sum _{d|n}{d}^{-1}.$

Abundant numbers are numbers with abundancy greater than 2, while perfect numbers are numbers with abundancy equal to 2 and deficient numbers are numbers with abundancy less than 2. When the abundancy of a positive integer n is a positive integer  

k n, k   ≥   1,

we have a k-perfect number, 1 being the only 1-perfect number.

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, ...

Properties

Any positive multiple of an abundant number is also an abundant number. Furthermore, any positive multiple (greater than 1) of a perfect number is an abundant number.

Theorem AbT1.

All positive multiples of abundant numbers are also abundant: Given a positive abundant number n and any positive integer m, the number  

m n

is also abundant.

Proof. It suffices to prove that  

n p

is abundant where  

n

is abundant and  

p

is prime, because  

m

is a product of zero or more primes and they can be applied by induction. If  

n

and  

p

are coprime, then  

σ−1(n p) = σ−1(n)  ⋅   σ−1( p) > 2  ⋅   σ−1( p) > 2

by the multiplicativity of  

σ−1

. Otherwise, let  

n = k p e

where  

k

and  

p

are coprime and note that  

σ−1(n p) = σ−1(k p e +1) = σ−1(k)  ⋅   σ−1( p e +1) > σ−1(k)  ⋅   σ−1( p e) = σ−1(k p e) = σ−1(n) = 2

since  

σ−1( p e)

is strictly increasing in  

e

. □

For example,  

12

is abundant, and so by the theorem its positive multiples are also abundant:  

{12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, ...}

(A008594).

Corollary AbC1.

All positive multiples  

m, m > 1

, of perfect numbers are abundant.

Proof. The above proof suffices, noting that  

σ−1( p) > 1

and  

σ−1( p e +1) > σ−1( p e)

. □

Corollary AbC2.

The abundant numbers have positive lower density.

Proof. 6 is a perfect number, so by Corollary AbC1 the lower density is at least  

1/6

. □

The abundant numbers have density of

• at least  

  1 6 = 0.166666666...

  (since 6 is perfect);
• at least  

  4 21 = 0.190476190...

  (since 6 and 28 are perfect);
• at least  

  23 105 = 0.219047619...

  (since 6 and 28 are perfect and 20 is primitive abundant).

Deléglise gives better bounds: their lower density is at least  

0.2474

and their upper density is at most  

0.2480

. All even numbers greater than 46 can be expressed as the sum of two abundant numbers in at least one way. For example,  

90 = 70 + 20

.

Theorem AbT2.

All even numbers greater than 46 are the sum of two abundant numbers in at least one way.

Proof. Recall that all multiples of an abundant number are also abundant, and that all multiples of a perfect number save the perfect number itself are abundant (by Theorem AbT1 above and its corollary). Now consider an even number  

n > 46

, but modulo  

12

, thus giving us just six cases to consider. Note that  

12

is an abundant number. If  

n   ≡   0 (mod 12)

, this means that  

n

is a multiple of  

12

and can be expressed as a sum of smaller multiples of  

12

in at least two ways (since  

n > 46

, e.g.,  

48 = 36 + 12 = 24 + 24

). Note that  

20

is an abundant number. If  

n   ≡   2 (mod 12)

, we can do  

n = 6m + 20

with an odd integer  

m > 1

(thus assuring  

6m

is abundant). If  

n   ≡   4 (mod 12)

, we can do  

n = 12m + 40

. If  

n   ≡   6 (mod 12)

, we can do  

n = 6m + 12

. If  

n   ≡   8 (mod 12)

, we can do  

n = 12m + 20.

We have purposely left  

n   ≡   10 (mod 12)

for last, since  

46   ≡   10 (mod 12)

. For this case, we can do  

n = 6m + 40

, in which both addends are abundant provided  

m > 1

. This exhausts all six cases, proving the theorem. □

Theorem AbT3.

All integers greater than 20161 are expressible as the sum of two abundant numbers in at least one way.

Proof. Following Parkin & Lander, write

$n=88e+315o$

where e is even and 2 < o < 90 is odd. 88e is abundant by Corollary AbC1 and it can be checked that 315o is also abundant (as $315=3^2\cdot 5\cdot 7$ it suffices to check o = 3, 7, and 89). This form can represent all odd n > 28122 and even numbers are handled by Theorem AbT2, so it suffices to check that the odd numbers between 20162 and 28122 can be expressed as such sums. □

A048242 Numbers that are not the sum of two abundant numbers (not necessarily distinct). ( 

a (38) = 46

is the largest even term;  

a (1456) = 20161

is the largest term.)

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 39, 41, 43, 45, 46, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, ..., 20161

Primitive abundant numbers

A091191 Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor (abundant numbers all of whose proper divisors are either deficient numbers or perfect numbers). (Union of A071395 and A275082.)

12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114, 138, 174, 186, 196, 222, 246, 258, 272, 282, 304, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 550, 572, 582, 606, 618, 642, 644, 650, 654, 678, 748, 762, 786, 812, 822, ...

A071395 Primitive abundant numbers (abundant numbers all of whose proper divisors are deficient numbers).

20, 70, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4030, 4070, 4095, 4216, 4288, ...

A275082 Primitive abundant numbers (having no abundant proper divisors) that have perfect proper divisors. (All even, since there are no known odd perfect numbers...)

12, 18, 30, 42, 56, 66, 78, 102, 114, 138, 174, 186, 196, 222, 246, 258, 282, 308, 318, 354, 364, 366, 402, 426, 438, 474, 476, 498, 532, 534, 582, 606, 618, 642, 644, 654, 678, 762, 786, 812, 822, 834, 868, 894, 906, 942, 978, 992, 1002, 1036, 1038, 1074, 1086, 1146, ...

A006038 Odd primitive abundant numbers (odd abundant numbers all of whose proper divisors are odd deficient numbers, since there are no known odd perfect numbers...).

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 7425, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, 28215, ...

Nonprimitive abundant numbers

A091192 Abundant numbers (A005101) having at least one abundant proper divisor.

24, 36, 40, 48, 54, 60, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 260, 264, 270, 276, 280, 288, 294, 300, 306, 312, 320, 324, 330, 336, 340, ...

A?????? Odd abundant numbers having at least one odd abundant proper divisor.

2835, 4725, 6615, 7875, 8505, 10395, 11025, 12285, 14175, ...

Apparently only contained in A005231, A174535, A174865 and A248694.

Other subsets

A173490 Even abundant numbers (even numbers  

n

whose sum of divisors exceeds  

2n

):

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, …

A?????? Even abundant numbers having no perfect proper divisor.

20, 40, 70, 80, 88, 100, 104, 140, 160, 176, 200, …

Cf. A064409, A093891, A177085, A192819, A204829, A280149 for potential supersets.

A005231 Odd abundant numbers (odd numbers  

n

whose sum of divisors exceeds  

2n

):

945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, …

A004490: Colossally abundant numbers:

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, …

A002093 Highly abundant numbers:  

σ (n) > σ (m), ∀m < n

.

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, …

A004394 Superabundant numbers: n such that  

σ (n) n > σ (m) m , ∀m < n

.

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, …

See also

• Pseudoperfect numbers
• Weird numbers
• Multiply-perfect numbers