Odd-abundant number

An abundant number is a number whose proper divisors sum to a value greater than itself. For example, 36 has proper divisors 1, 2, 3, 4, 6, 9, 12, 18, which sum to 55, and \(55 > 36\). Contrast deficient numbers, whose proper divisors sum to a smaller value, and perfect numbers, whose proper divisors sum to themselves.
The first few abundant numbers are 12, 18, 20, 24, 30, 36, 40, 42, 48, ... Note that most of these appear to be even; the first odd one does not appear until 945. This is quite remarkable, as it provides a naturally occurring example of a large number. After 945, the odd-abundant numbers are 1,575, 2,205, 2,835, 3,465, ...OIES A005231
There are also abundant numbers whose proper divisors have a sum greater than twice the original number. The smallest one is 180, but no odd ones occur until 1,018,976,683,725.
== Notable examples ==
These are special odd-abundant numbers, such as:
*945 is the smallest odd-abundant number.Abundant numbers on Wikipedia
**945 = 33*5*7, so the sum of all divisors of 945 (including itself) is (1+3+32+33)*(1+5)*(1+7) = 40*6*8 = 1,920, while 945*2=1,890<1,920. Therefore, 945 is an abundant number. By trial and error we can check it is also smallest odd number with this property.
*1,575 is the smallest odd-abundant partition number, and the smallest abundant number with only odd digits.http://www.numbersaplenty.com/both_abundant_and_partition.html
*2,205 is the smallest heptagonal odd-abundant number.http://www.numbersaplenty.com/both_abundant_and_heptagonal.html
**The lasted from to , for a total of 2,205 years.
**For this reason, a video game has been named .
**It is also approximately the number of in a .
*2,835 is the smallest decagonal odd-abundant number.http://www.numbersaplenty.com/both_abundant_and_decagonal.html
*3,465 is named “obragsracx” by André Joyce.
*4,095 is the smallest triangular odd-abundant numberhttp://www.numbersaplenty.com/both_abundant_and_triangular.html and also the smallest odd abundant Mersenne number.
*5,355 is the smallest abundant number using only digits 3 and 5.
*5,775 is the smallest palindromic odd-abundant number, the smallest odd-abundant number with abundant successor, and the smallest abundant number using only digits 5 and 7.http://www.numbersaplenty.com/both_abundant_and_palindromic.html
*5,985 is the smallest octagonal odd-abundant number, and the smallest odd-abundant number with abundant predecessor.http://www.numbersaplenty.com/both_abundant_and_octagonal.html
*7,875 is the smallest hexagonal odd-abundant number.http://www.numbersaplenty.com/both_abundant_and_hexagonal.html
*9,555 is the smallest abundant number using only digits 5 and 9.
*11,025 is the smallest square odd-abundant number.http://www.numbersaplenty.com/both_abundant_and_square.html
*12,285 is the smallest odd-abundant number which is part of an amicable pair (namely, with 14,595).
*20,475 is the smallest pentagonal odd-abundant number.http://www.numbersaplenty.com/both_abundant_and_pentagonal.html
*42,075 is the smallest nonagonal odd-abundant number.http://www.numbersaplenty.com/both_abundant_and_nonagonal.html
*50,505 is the smallest odd-abundant undulating number.http://www.numbersaplenty.com/both_abundant_and_undulating.html
*81,081 is the smallest abundant number ending in the digit 1.
*400,995 is the smallest tetrahedral odd-abundant number.http://www.numbersaplenty.com/both_abundant_and_tetrahedral.html
*555,555 is the smallest odd-abundant repdigit.http://www.numbersaplenty.com/both_abundant_and_repdigit.html
*1,157,625 is the smallest cube odd-abundant number.http://www.numbersaplenty.com/both_abundant_and_cube.html
*9,694,845 is the smallest odd-abundant Catalan number.http://www.numbersaplenty.com/both_abundant_and_Catalan.html
*10,173,345 is the smallest odd-abundant magic constant.http://www.numbersaplenty.com/both_abundant_and_magic.html
*19,571,895 is the smallest odd-abundant house number.http://www.numbersaplenty.com/both_abundant_and_house.html
*28,158,165 is the smallest odd-abundant number which is part of an aliquot cycle of length 4.http://djm.cc/sociable.txt
*171,078,831 is the smallest odd-abundant number with abundant predecessor and abundant successor.
**Its prime factorisation is \(5^2*7*11*13*17*19*23*29\).
*333,333,333,333 is the smallest abundant number containing only threes.http://www.numbersaplenty.com/333333333333
*111,111,111,111,111,111 is the smallest abundant repunit.
**The prime factorization of this number is:
**7 x 11 x 13 x 17 x 19 x 23 x 29 x 31 x 37 x 41 x 43 x 47 x 53 x 59 x 61 x 67
*7,970,466,327,524,571,538,225,709,545,434,506,255,970,026,969,710,012,787,303,278,390,616,918,473,506,860,039,424,701 is the smallest abundant number that is not a multiple of primes less than 13.Odd abundant numbers - OeisWiki
== Sources ==
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