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Term

Deficient Number

Numbers whose proper divisors sum to less than the number itself — the most common kind.

10,857 numbers tagged.

A positive integer is deficient when the sum of its proper divisors is less than the number itself. Most integers are deficient — every prime is deficient (its only proper divisor is 1), as is every prime power and many composites.

The number 10's proper divisors are 1, 2, 5 — summing to 8, less than 10. So 10 is deficient. The deficiency \(n - \sigma_0(n)\) measures how far a number is from being perfect.

Numbers fall into exactly three categories based on their proper-divisor sum: deficient, perfect, or abundant. Of these, deficient numbers are by far the most common — about 75% of integers are deficient.

22,727 22,739 22,741 22,751 22,769 22,777 22,783 22,787 22,807 22,811 22,817 22,853 22,859 22,861 22,871 22,877 22,901 22,907 22,921 22,937 22,943 22,961 22,963 22,973 22,993 23,003 23,011 23,017 23,021 23,027 23,029 23,039 23,041 23,053 23,057 23,059 23,063 23,071 23,081 23,087 23,099 23,117 23,131 23,143 23,159 23,167 23,173 23,189 23,197 23,201 23,203 23,209 23,227 23,251 23,269 23,279 23,291 23,293 23,297 23,311 23,321 23,327 23,333 23,339 23,357 23,369 23,371 23,399 23,417 23,431 23,447 23,459 23,473 23,497 23,509 23,531 23,537 23,539 23,549 23,557 23,561 23,563 23,567 23,581 23,593 23,599 23,603 23,609 23,623 23,627 23,629 23,633 23,663 23,669 23,671 23,677 23,687 23,689 23,719 23,741 23,743 23,747 23,753 23,761 23,767 23,773 23,789 23,801 23,813 23,819 23,827 23,831 23,833 23,857 23,869 23,873 23,879 23,887 23,893 23,899 23,909 23,911 23,917 23,929 23,957 23,971 23,977 23,981 23,993 24,001 24,007 24,019 24,023 24,029 24,043 24,049 24,061 24,071 24,077 24,083 24,091 24,097 24,103 24,107 24,109 24,113 24,121 24,133 24,137 24,151 24,169 24,179 24,181 24,197 24,203 24,223 24,229 24,239 24,247 24,251 24,281 24,317 24,329 24,337 24,359 24,371 24,373 24,379 24,391 24,407 24,413 24,419 24,421 24,439 24,443 24,469 24,473 24,481 24,499 24,509 24,517 24,527 24,533 24,547 24,551 24,571 24,593 24,611 24,623 24,631 24,659 24,671 24,677 24,683 24,691 24,697 24,709 24,733 24,749 24,763