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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
12 = 2 2 ⋅ 3, with
σ (12) = ⋅ (3 + 1) = 7 ⋅ 4 = 28 > 24 = 2 ⋅ 12 
, the first
odd abundant number (which happens to be the
232nd 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 
.
Arithmetic sequences yielding many odd abundant numbers
The formula^{[1]}

a (n) = 3 ⋅ 105 ⋅ (3 + 2 n) = 3 ⋅ (315 + 210 n) = 945 + 630 n = A005231 (1) + 3 ⋅ p4 # ⋅ n, 0 ≤ n ≤ 51, 
where
is the
th primorial number, gives
52 odd abundant numbers, but fails to give an abundant number for
.
The formula^{[2]}

a (n) = 11 ⋅ 105 ⋅ (3 + 2 n) = 11 ⋅ (315 + 210 n) = 3465 + 2310 n = A005231 (5) + p5 # ⋅ n, 0 ≤ n ≤ 192, 
where
is the
th primorial number, gives
193 odd abundant numbers, but fails to give an abundant number for
.
Infinitude of odd abundant numbers
There are infinitely many odd [nonprimitive] abundant numbers: Odd abundant numbers are closed under multiplication by arbitrary positive odd integers, since any positive multiple of a [primitive or nonprimitive] abundant number is [nonprimitive] abundant.
Are there infinitely many odd [primitive] abundant numbers?: Conjecture or proof? ^{(Provide proof: PROOF GOES HERE. □)} ^{[3]}
Odd primitive abundant numbers
A006038 Odd primitive abundant 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, ...}
A?????? Odd nonprimitive abundant numbers. ^{ (Add to OEIS?: Odd nonprimitive abundant numbers.)} ^{[4]}

{2835, 4725, 6615, 7875, 8505, 10395, 11025, 12285, 14175, 15435, 16065, 17325, 17955, ...}
Avoiding other prime factors
Avoiding a single prime factor p
The first odd abundant number is
 945 = 3 3 ⋅ 5 ⋅ 7, this being the 1st odd abundant number and the 232nd abundant number.
The first odd abundant number not divisible by 3 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 5 is
 81081 = 3 4 ⋅ 7 ⋅ 11 ⋅ 13, this being the 175th odd abundant number and the ?th abundant number.
The first odd abundant number not divisible by 7 is
 6435 = 3 2 ⋅ 5 ⋅ 11 ⋅ 13, this being the 11th odd abundant number and the 1601st abundant number.
A?????? Smallest odd abundant number which is not divisible by the
th odd prime.
^{ (Add to OEIS?)} ^{[5]}

{5391411025, 81081, 6435, ...}
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 odd
.
^{ (Add to OEIS?)} ^{[6]}

{945, 945, 5391411025, 945, 81081, 5391411025, ...}
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 3) 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
1st odd abundant number and the
232nd abundant number.
The first 5rough (i.e. not divisible by primes less than 5) 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 7) 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 11) 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 13) 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,
. (Smallest abundant number
s.t.
, for
.)

{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? Is the asymptotic density of odd abundant numbers among the abundant numbers equal to , although maybe with a bias in favor of even abundant numbers?
Notes