Aliquot sum

<h2 id="examples">Examples</h2>
For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are 1, 2, 3, 4, and 6, so the aliquot sum of 12 is 16 i.e. (1 + 2 + 3 + 4 + 6).
The values of s(n) for n = 1, 2, 3, ... are:

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, ... (sequence A001065 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>)
<h2 id="characterization-of-classes-of-numbers">Characterization of classes of numbers</h2>
The aliquot sum function can be used to characterize several notable classes of numbers:

<ul><li>1 is the only number whose aliquot sum is 0.</li>
<li>A number is <a href="/facts/Prime_number/L20FLkgn">prime</a> if and only if its aliquot sum is 1.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li>
<li>The aliquot sums of <a href="/facts/Perfect_number/WyqWtApV">perfect</a>, <a href="/facts/Deficient_number/dreFgv8A">deficient</a>, and <a href="/facts/Abundant_number/K54Q9owk">abundant</a> numbers are equal to, less than, and greater than the number itself respectively.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> The <a href="/facts/Quasiperfect_number/3BCbnTUC">quasiperfect numbers</a> (if such numbers exist) are the numbers n whose aliquot sums equal n + 1. The <a href="/facts/Almost_perfect_number/7pzWByG0">almost perfect numbers</a> (which include the powers of 2, being the only known such numbers so far) are the numbers n whose aliquot sums equal n – 1.</li>
<li>The <a href="/facts/Untouchable_number/RWtqyP39">untouchable numbers</a> are the numbers that are not the aliquot sum of any other number. Their study goes back at least to <a href="/facts/Abu_Mansur_al-Baghdadi/e1PmYTd9">Abu Mansur al-Baghdadi</a> (circa 1000 AD), who observed that both 2 and 5 are untouchable.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> <a href="/facts/Paul_Erd%25C5%2591s/63LoUTe1">Paul Erdős</a> proved that their number is infinite.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> The conjecture that 5 is the only odd untouchable number remains unproven, but would follow from a form of <a href="/facts/Goldbach%2527s_conjecture/967UhtUU">Goldbach's conjecture</a> together with the observation that, for a <a href="/facts/Semiprime_number/J5cNFs7T">semiprime number</a> pq, the aliquot sum is p + q + 1.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li></ul>
The mathematicians Pollack & Pomerance (2016) noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.

<h2 id="iteration">Iteration</h2>
Main article: <a href="/facts/Aliquot_sequence/XYI13Wne">Aliquot sequence</a>
<a href="/facts/Iterated_function/Vzsx64An">Iterating</a> the aliquot sum function produces the <a href="/facts/Aliquot_sequence/XYI13Wne">aliquot sequence</a> n, s(n), s(s(n)), … of a nonnegative integer n (in this sequence, we define s(0) = 0). 
<a href="/facts/Sociable_number/1typvFxR">Sociable numbers</a> are numbers whose aliquot sequence is a <a href="/facts/Periodic_sequence/eIJhDfJn">periodic sequence</a>. <a href="/facts/Amicable_numbers/Rx7QgbXI">Amicable numbers</a> are sociable numbers whose aliquot sequence has period 2.
It remains unknown whether these sequences always end with a <a href="/facts/Prime_number/L20FLkgn">prime number</a>, a <a href="/facts/Perfect_number/WyqWtApV">perfect number</a>, or a periodic sequence of sociable numbers.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Divisor_function/rzvm1jw2">Sum of positive divisors function</a>, the sum of the (xth powers of the) positive divisors of a number</li>
<li><a href="/facts/William_of_Auberive/E7Y4CGtj">William of Auberive</a>, medieval numerologist interested in aliquot sums</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/RestrictedDivisorFunction.html">"Restricted Divisor Function"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Pollack, Paul; Pomerance, Carl (2016), "Some problems of Erdős on the sum-of-divisors function", Transactions of the American Mathematical Society, Series B, 3: 1–26, doi:10.1090/btran/10, MR 3481968 <a href="/wiki/Carl_Pomerance" target="_blank">/wiki/Carl_Pomerance</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Pollack, Paul; Pomerance, Carl (2016), "Some problems of Erdős on the sum-of-divisors function", Transactions of the American Mathematical Society, Series B, 3: 1–26, doi:10.1090/btran/10, MR 3481968 <a href="/wiki/Carl_Pomerance" target="_blank">/wiki/Carl_Pomerance</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Pollack, Paul; Pomerance, Carl (2016), "Some problems of Erdős on the sum-of-divisors function", Transactions of the American Mathematical Society, Series B, 3: 1–26, doi:10.1090/btran/10, MR 3481968 <a href="/wiki/Carl_Pomerance" target="_blank">/wiki/Carl_Pomerance</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Sesiano, J. (1991), "Two problems of number theory in Islamic times", Archive for History of Exact Sciences, 41 (3): 235–238, doi:10.1007/BF00348408, JSTOR 41133889, MR 1107382, S2CID 115235810 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Erdős, P. (1973), "Über die Zahlen der Form 
 
 
 
 σ
 (
 n
 )
 −
 n
 
 
 {\displaystyle \sigma (n)-n}
 
 und 
 
 
 
 n
 −
 ϕ
 (
 n
 )
 
 
 {\displaystyle n-\phi (n)}
 
" (PDF), Elemente der Mathematik, 28: 83–86, MR 0337733 <a href="/wiki/Paul_Erd%C5%91s" target="_blank">/wiki/Paul_Erd%C5%91s</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Pollack, Paul; Pomerance, Carl (2016), "Some problems of Erdős on the sum-of-divisors function", Transactions of the American Mathematical Society, Series B, 3: 1–26, doi:10.1090/btran/10, MR 3481968 <a href="/wiki/Carl_Pomerance" target="_blank">/wiki/Carl_Pomerance</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Weisstein, Eric W. "Catalan's Aliquot Sequence Conjecture". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Aliquot sum open-in-new

Aliquot sum