In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 10 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 14. The amount by which the sum exceeds the number is the abundance. The number 10 has an abundance of 4, for example.

## Definition

A number n for which the sum of divisors σ(n)>2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)>n.

Abundance is the value σ(n)-2n (or s(n)-n).

## Mod n

Pictorial representation of remainders (mod 1, 2, 3, ...,10) frequency for such numbers ≤ 106:

modulo
1 2E7859
2 2E42EX 355E
3 20355X 58168 58153
4 1X3654 1883 110866 1898
5 XXX60 62253 62250 62266 62270
6 1EEEEE 0 58153 355E 58168 0
7 7992X 477E5 47795 477E0 477X5 47799 477XE
8 1040EE X40 66436 X60 9E555 X43 66430 X38
9 828X0 1X866 1X856 80444 1X85X 1X856 80436 1X864 1X863
X X7582 23 62233 21 62250 349X 62230 19 62245 20
E 41851 31976 31958 31970 31973 31993 31965 3197X 31967 31970 31982
10 100000 0 642X 1898 5192E 0 EEEEE 0 51925 1883 6439 0
remainder 0 1 2 3 4 5 6 7 8 9 X E

A list for how many such numbers ≤ 106 are multiples of the numbers from 1 to 100.

+1 +2 +3 +4 +5 +6 +7 +8 +9 +X +E +10
0+ 2E7859 2E42EX 20355X 1X3654 XXX60 1EEEEE 7992X 1040EE 828X0 X7582 41851 100000
10+ 35139 76E4X 51002 674XE 25227 80000 21E79 72497 37E66 4075X 18E20 60000
20+ 23772 343E7 2937X 51869 144X5 49724 1338E 35913 23315 24822 23547 40000
30+ 10447 2168E 1XE23 37249 E154 35186 X709 280EX 19X68 18773 97E6 30000
40+ 11767 22645 1553E 22807 866E 28000 126E0 26X35 13696 1423X 77X1 24972
50+ 7492 13150 13X22 196X6 EX75 22222 682X 16X4E 1099X 20828 6310 20000
60+ 60E2 10269 10757 14995 9286 1X1X1 5711 19724 E141 XE99 5389 186X3
70+ 8594 X567 X187 17777 4E37 17249 78E0 11600 9598 9672 7644 16000
80+ 4643 11088 9540 15343 4419 14E36 4311 14747 XEX9 8551 411X 14000
90+ 402E 11674 7E28 13518 3X63 131X8 6138 X697 7X93 769E 592X 12497
X0+ 4725 7397 71E1 9X37 5673 1186X 3536 XX32 69X7 E193 3402 11111
E0+ 5162 6746 7800 E258 3226 10631 317X 10414 62X8 6238 4XX4 10000