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# Odd abundant numbers

(Redirected from Odd abundant number)

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
.

## Arithmetic sequences yielding many odd abundant numbers

The formula

 a (n) = 3 ⋅  105 ⋅  (3 + 2 n) = 3 ⋅  (315 + 210 n) = 945 + 630 n = A005231 (1) + 3 ⋅  p4 # ⋅  n, 0 ≤ n ≤ 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

 a (n) = 11 ⋅  105 ⋅  (3 + 2 n) = 11 ⋅  (315 + 210 n) = 3465 + 2310 n = A005231 (5) + p5 # ⋅  n, 0 ≤ n ≤ 192,
where
 pn #
is the
 n
th primorial number, gives 193 odd abundant numbers, but fails to give an abundant number for
 n = 193
.

## 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. □)

### 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..)

{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
 n
th odd prime.  (Add to OEIS?.)
{5391411025, 81081, 6435, ...}
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 odd
 k = 2 n + 1, n   ≥   1
{945, 945, 5391411025, 945, 81081, 5391411025, ...}

### 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 (k ) > 2 k
) which is not divisible by any of the first
 n
primes,
 n   ≥   0
. (Smallest abundant number
 k
s.t.
 gcd (σ (k ), pn # )   ≡   1
, for
 n   ≥   0
.)
{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? Is the asymptotic density of odd abundant numbers among the abundant numbers equal to
 1 2
, although maybe with a bias in favor of even abundant numbers?