11,051 Pages

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, ...

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.
• 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 odd digits.
• 2,205 is the smallest heptagonal odd-abundant number.
• 2,835 is the smallest decagonal odd-abundant number.
• 3,465 is named “obragsracx” by André Joyce.
• 4,095 is the smallest triangular odd-abundant number 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.
• 5,985 is the smallest octagonal odd-abundant number, and the smallest odd-abundant number with abundant predecessor.
• 7,875 is the smallest hexagonal odd-abundant number.
• 9,555 is the smallest abundant number using only digits 5 and 9.
• 11,025 is the smallest square odd-abundant number.
• 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.
• 42,075 is the smallest nonagonal odd-abundant number.
• 50,505 is the smallest odd-abundant undulating number.
• 81,081 is the smallest abundant number ending in the digit 1.
• 151,515 is the smallest abundant number using only digits 1 and 5.
• 153,153 is the smallest abundant number ending in the digit 3.
• 171,171 is the smallest odd-abundant iban number, the smallest abundant number using only odd digits not equal to 5, and the smallest abundant number using only digits 1 and 7.
• 189,189 is the smallest abundant number ending in the digit 9.
• 207,207 is the smallest abundant number ending in the digit 7, and also an iban number.
• 243,243 is an odd-abundant iban number.
• 400,995 is the smallest tetrahedral odd-abundant number.
• 555,555 is the smallest odd-abundant repdigit.
• 742,203 is the largest odd-abundant iban number.
• 999,999 is the smallest abundant number containing only nines.
• 1,157,625 is the smallest cube odd-abundant number.
• 9,694,845 is the smallest odd-abundant Catalan number.
• 10,173,345 is the smallest odd-abundant magic constant.
• 19,571,895 is the smallest odd-abundant house number.
• 28,158,165 is the smallest odd-abundant number which is part of an aliquot cycle of length 4.
• 171,078,831 is the smallest odd-abundant number with abundant predecessor and abundant successor.
• 5,391,411,025 is the smallest abundant number not divisible by neither 2 nor 3. It is also the smallest odd-abundant number not divisible by 3.
• Its prime factorisation is $$5^2*7*11*13*17*19*23*29$$ 
• 21,548,919,483 is the smallest odd-abundant number which is part of an aliquot cycle of length 6.
• 333,333,333,333 is the smallest abundant number containing only threes.
• 1,018,976,683,725 is the smallest odd-abundant number whose factors have a sum greater than three times the original number.
• 138,344,559,911,415 is the smallest odd-abundant number which is part of an aliquot cycle of length 8.
• 111,111,111,111,111,111 is the smallest abundant repunit.
• 777,777,777,777,777,777 is the smallest abundant number containing only sevens.
• 20,821,017,304,425,168,561,312,837,502,762,890,375 is the first odd-abundant number whose factors has a sum of more than 4 times the original number. It is approximately equal to $$2.082 \times 10^{37}$$.
• The prime factorization of this number is:
• 35 × 53 × 72 × 112 × 13 × 17 × 19 × 23 × 29 × 31 × 37 × 41 × 43 × 47 × 53 × 59 × 61 × 67 × 71 × 73 × 79 × 83
• 48,870,871,124,826,570,463,953,805,139,878,697,155,358,000,962,012,333,290,725,030,523,875 is the first odd-abundant number not divisible by 3 and whose factors has a sum of more than 3 times the original number. It is approximately equal to $$4.887 \times 10^{67}$$.
• The prime factorization of this number is:
• 53 x 73 x 112 x 132 x 172 x 192 x 232 x 29 x 31 x 37 x 41 x 43 x 47 x 53 x 59 x 61 x 67 x 71 x 73 x 79 x 83 x 89 x 97 x 101 x 103 x 107 x 109 x 113 x 127 x 131 x 137 x 139 x 149 x 151.
• Therefore, the sum of the factors is:
• (1+5+52+53) * (1+7+72+73) * (1+11+112) * (1+13+132) * (1+17+172) * (1+19+192) * (1+23+232) * 30 * 32 * 38 * 42 * 44 * 48 * 54 * 60 * 62 * 68 * 72 * 74 * 80 * 84 * 90 * 98 * 102 * 104 * 108 * 110 * 114 * 128 * 132 * 138 * 140 * 150 * 152 = 146,623,032,169,592,356,083,568,880,299,404,643,621,639,498,625,919,913,820,160,000,000,000.
• This is 3.000213190288... times the original number.
• 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.

## Sources

Community content is available under CC-BY-SA unless otherwise noted.