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# Odd abundant numbers

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

## Odd abundant numbers arithmetic sequences

The formula[1]

${\displaystyle {a(n)=3\cdot 105\cdot (3+2n)=3\cdot (315+210n)=945+630n=A005231(1)+3\cdot {p_{4}}\#\cdot n,\quad 0\leq n\leq 51},\,}$
where
 pn #
is the
 n
th primorial number, gives
 52
odd abundant numbers, but fails to give an abundant number for
 n = 52
.

The formula[2]

${\displaystyle {a(n)=11\cdot 105\cdot (3+2n)=11\cdot (315+210n)=3465+2310n=A005231(5)+{p_{5}}\#\cdot n},\quad 0\leq n\leq 192,\,}$
where
 pn #
is the
 n
th primorial number, gives
 193
odd abundant numbers, but fails to give an abundant number for
 n = 193
.

## 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

 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.
A114371 Smallest abundant number relatively prime to
 n, n   ≥   1
.
 {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
 n, n   ≥   1
.
 {945, 945, 5391411025, 945, 81081, 5391411025, ...}
A?????? Smallest odd abundant number which is not divisible by the
 n
th prime.
 {945, 5391411025, 81081, ...}

### Avoiding all prime factors up to p

In the following,
 ( pn) #
is the
 n
th primorial number.
The first 3-rough (i.e. not divisible by primes less than
 3
) abundant number is
945 = 3 3  ⋅   5  ⋅   7 = 3 2  ⋅
 ( p4) # ( p1) #
,
with 3 distinct prime factors, those being the first 4 primes excluding the first prime. We have
 σ (945) = 1920
, abundance of
 1920  −  1890 = 30
, abundancy of
 1920 1890
=
 64 63
, this being the 1st odd abundant number and the 232nd abundant number.
The first 5-rough (i.e. not divisible by primes less than
 5
) abundant number is
5391411025 = 5 2  ⋅   7  ⋅   11  ⋅   13  ⋅   17  ⋅   19  ⋅   23  ⋅   29 = 5  ⋅
 ( p10) # ( p2) #
,
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
 10799308800 10782822050
=
 30855168 30808063
, this being the ?th odd abundant number and the ?th abundant number.
The first 7-rough (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  ⋅
 ( p19) # ( p3) #
,

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 11-rough (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  ⋅
 ( p33) # ( p4) #
,

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 13-rough (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  ⋅
 ( p49) # ( p5) #
,

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 (
 σ (n) > 2n
) which is not divisible by any of the first
 n
primes.
 {12, 945, 5391411025, 20169691981106018776756331, 49061132957714428902152118459264865645885092682687973, 7970466327524571538225709545434506255970026969710012787303278390616918473506860039424701, ...}
A107705
 a (n)
is the least number of prime factors in any non-deficient number that has the
 n
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
 a (n)
is the least number of prime factors for any abundant number with
 pn
(the
 n
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??????
 a (n)
is the least number of distinct prime factors for any abundant number with
 pn
(the
 n
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

1. Jay L. Schiffman, Odd Abundant Numbers, Math. Spectrum 37 (2004/2005), pp. 73–75.
2. Jay L. Schiffman and Christopher S. Simons, More Odd Abundant Sequences, 2005.