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The odd abundant numbers are odd numbers which are abundant, i.e. whose sum of divisors is greater than twice the number (or whose sum of aliquot divisors is greater than the number).
A005231 Odd abundant numbers.

{945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9555, 9765, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175, ...} 
While the first
even abundant number is
, with
σ (12) = ⋅ (3 + 1) = 7 ⋅ 4 = 28 > 24 = 2 ⋅ 12 
, the first
odd abundant number (which happens to be 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) = ⋅ (5 + 1) ⋅ (7 + 1) = 40 ⋅ 6 ⋅ 8 = 1920 > 1890 = 2 ⋅ 945 
.
Odd abundant numbers arithmetic sequences
The formula^{[1]}
 ${a(n)=3\cdot 105\cdot (3+2n)=3\cdot (315+210n)=945+630n=A005231(1)+3\cdot {p_{4}}\#\cdot n,\quad 0\leq n\leq 51},\,$
where
is the
th primorial number, gives
odd abundant numbers, but fails to give an abundant number for
.
The formula^{[2]}
 ${a(n)=11\cdot 105\cdot (3+2n)=11\cdot (315+210n)=3465+2310n=A005231(5)+{p_{5}}\#\cdot n},\quad 0\leq n\leq 192,\,$
where
is the
th primorial number, gives
odd abundant numbers, but fails to give an abundant number for
.
Properties
Odd abundant numbers are closed under multiplication by arbitrary positive odd integers, since any positive multiple of an abundant number is abundant. There are thus infinitely many odd abundant numbers.
Avoiding other prime factors
Avoiding a single prime factor p
The first odd abundant number is
 , this being the 1^{st} odd abundant number and the 232^{nd} abundant number.
The first odd abundant number not divisible by
is

5391411025 = 5 2 ⋅ 7 ⋅ 11 ⋅ 13 ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 29 
, this being the ?th odd abundant number and the ?th abundant number.
The first odd abundant number not divisible by
is

81081 = 3 4 ⋅ 7 ⋅ 11 ⋅ 13 
, this being the 175^{th} odd abundant number and the ?th abundant number.
The first odd abundant number not divisible by
is
 , this being the 11^{th} odd abundant number and the 1601^{st} abundant number.
A114371 Smallest abundant number relatively prime to
.

{12, 945, 20, 945, 12, 5391411025, 12, 945, 20, 81081, 12, 5391411025, 12, 6435, 56, 945, 12, 5391411025, 12, 81081, 20, 945, 12, 5391411025, 12, 945, 20, 6435, 12, ...} 
A?????? Smallest odd abundant number relatively prime to
.

{945, 945, 5391411025, 945, 81081, 5391411025, ...} 
A?????? Smallest odd abundant number which is not divisible by the
th prime.

{945, 5391411025, 81081, ...} 
Avoiding all prime factors up to p
In the following,
is the
th primorial number.
The first
3rough (i.e. not divisible by primes less than
) abundant number is

945 = 3 3 ⋅ 5 ⋅ 7 = 3 2 ⋅ , 
with 3
distinct prime factors, those being the first 4 primes excluding the first prime. We have
,
abundance of
,
abundancy of
, this being the 1
^{st} odd abundant number and the 232
^{nd} abundant number.
The first
5rough (i.e. not divisible by primes less than
) abundant number is

5391411025 = 5 2 ⋅ 7 ⋅ 11 ⋅ 13 ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 29 = 5 ⋅ , 
with 8 distinct prime factors, those being the first 10 primes excluding the first 2 primes. We have
σ (5391411025) = 10799308800 
,
abundance of
10799308800 − 10782822050 = 16486750 
,
abundancy of
, this being the ?
th odd abundant number and the ?
th abundant number.
The first
7rough (i.e. not divisible by primes less than
) abundant number is

20169691981106018776756331 = 7 2 ⋅ 11 2 ⋅ 13 ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 29 ⋅ 31 ⋅ 37 ⋅ 41 ⋅ 43 ⋅ 47 ⋅ 53 ⋅ 59 ⋅ 61 ⋅ 67 = 7 ⋅ 11 ⋅ , 
with 16 distinct prime factors, those being the first 19 primes excluding the first 3 primes, this being the ?th odd abundant number and the ?th abundant number.
The first
11rough (i.e. not divisible by primes less than
) abundant number is

49061132957714428902152118459264865645885092682687973 = 11 2 ⋅ 13 2 ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 29 ⋅ 31 ⋅ 37 ⋅ 41 ⋅ 43 ⋅ 47 ⋅ 53 ⋅ 59 ⋅ 61 ⋅ 67 ⋅ 71 ⋅ 73 ⋅ 79 ⋅ 83 ⋅ 89 ⋅ 97 ⋅ 101 ⋅ 103 ⋅ 107 ⋅ 109 ⋅ 113 ⋅ 127 ⋅ 131 ⋅ 137 = 11 ⋅ 13 ⋅ , 
with 29 distinct prime factors, those being the first 33 primes excluding the first 4 primes, this being the ?th odd abundant number and the ?th abundant number.
The first
13rough (i.e. not divisible by primes less than
) abundant number is

7970466327524571538225709545434506255970026969710012787303278390616918473506860039424701 = 13 2 ⋅ 17 2 ⋅ 19 ⋅ 23 ⋅ 29 ⋅ 31 ⋅ 37 ⋅ 41 ⋅ 43 ⋅ 47 ⋅ 53 ⋅ 59 ⋅ 61 ⋅ 67 ⋅ 71 ⋅ 73 ⋅ 79 ⋅ 83 ⋅ 89 ⋅ 97 ⋅ 101 ⋅ 103 ⋅ 107 ⋅ 109 ⋅ 113 ⋅ 127 ⋅ 131 ⋅ 137 ⋅ 139 ⋅ 149 ⋅ 151 ⋅ 157 ⋅ 163 ⋅ 167 ⋅ 173 ⋅ 179 ⋅ 181 ⋅ 191 ⋅ 193 ⋅ 197 ⋅ 199 ⋅ 211 ⋅ 223 ⋅ 227 = 13 ⋅ 17 ⋅ , 
with 44 distinct prime factors, those being the first 49 primes excluding the first 5 primes, this being the ?th odd abundant number and the ?th abundant number.
Sequences
A047802 Smallest abundant number (
) which is not divisible by any of the first
primes.

{12, 945, 5391411025, 20169691981106018776756331, 49061132957714428902152118459264865645885092682687973, 7970466327524571538225709545434506255970026969710012787303278390616918473506860039424701, ...} 
A107705 is the least number of prime factors in any nondeficient number that has the
th prime as its least prime factor.
(If there are no odd perfect numbers, then only the first term would differ from A108227...)

{2, 5, 9, 18, 31, 46, 67, 91, 122, 157, 194, 238, 284, 334, 392, 455, 522, 591, 668, 748, 834, 929, 1028, 1133, 1241, 1352, 1469, 1594, 1727, 1869, 2019, 2163, 2315, 2471, 2636, ...} 
A108227 is the least number of prime factors for any abundant number with
(the
th prime) as its least factor.

{3, 5, 9, 18, 31, 46, 67, 91, 122, 157, 194, 238, 284, 334, 392, 455, 522, 591, 668, 748, 834, 929, 1028, 1133, 1241, 1352, 1469, 1594, 1727, 1869, 2019, 2163, 2315, 2471, 2636, ...} 
A??????
is the least number of distinct prime factors for any abundant number with
(the
th prime) as its least factor.

{2, 3, 8, 16, 29, 44, ...} 
Asymptotic density of odd abundant numbers among the abundant numbers
What is the asymptotic density of odd abundant numbers among the abundant numbers?
Notes