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Divisor function
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<h2 id="definition">Definition</h2> <p>The sum of positive divisors function <i>σ</i><i>z</i>(<i>n</i>), for a real or complex number <i>z</i>, is defined as the <a href="/facts/Summation/V2WUxoKn">sum</a> of the <i>z</i>th <a href="/facts/Exponentiation/pac6QY2g">powers</a> of the positive <a href="/facts/Divisor/DKNa2GvC">divisors</a> of <i>n</i>. It can be expressed in <a href="/facts/Summation/V2WUxoKn">sigma notation</a> as </p> σ z ( n ) = ∑ d ∣ n d z , {\displaystyle \sigma _{z}(n)=\sum _{d\mid n}d^{z}\,\!,} <p>where d ∣ n {\displaystyle {d\mid n}} is shorthand for "<i>d</i> <a href="/facts/Divides/DKNa2GvC">divides</a> <i>n</i>". The notations <i>d</i>(<i>n</i>), <i>ν</i>(<i>n</i>) and <i>τ</i>(<i>n</i>) (for the German <i>Teiler</i> = divisors) are also used to denote <i>σ</i>0(<i>n</i>), or the number-of-divisors function<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> (<a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A000005). When <i>z</i> is 1, the function is called the sigma function or sum-of-divisors function,<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> and the subscript is often omitted, so <i>σ</i>(<i>n</i>) is the same as <i>σ</i>1(<i>n</i>) (<a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A000203). </p><p>The <a href="/facts/Aliquot_sum/v3MQjbIU">aliquot sum</a> <i>s</i>(<i>n</i>) of <i>n</i> is the sum of the <a href="/facts/Proper_divisor/DKNa2GvC">proper divisors</a> (that is, the divisors excluding <i>n</i> itself, <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A001065), and equals <i>σ</i>1(<i>n</i>) − <i>n</i>; the <a href="/facts/Aliquot_sequence/XYI13Wne">aliquot sequence</a> of <i>n</i> is formed by repeatedly applying the aliquot sum function. </p> <h2 id="example">Example</h2> <p>For example, <i>σ</i>0(12) is the number of the divisors of 12: </p> σ 0 ( 12 ) = 1 0 + 2 0 + 3 0 + 4 0 + 6 0 + 12 0 = 1 + 1 + 1 + 1 + 1 + 1 = 6 , {\displaystyle {\begin{aligned}\sigma _{0}(12)&=1^{0}+2^{0}+3^{0}+4^{0}+6^{0}+12^{0}\\&=1+1+1+1+1+1=6,\end{aligned}}} <p>while <i>σ</i>1(12) is the sum of all the divisors: </p> σ 1 ( 12 ) = 1 1 + 2 1 + 3 1 + 4 1 + 6 1 + 12 1 = 1 + 2 + 3 + 4 + 6 + 12 = 28 , {\displaystyle {\begin{aligned}\sigma _{1}(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}+12^{1}\\&=1+2+3+4+6+12=28,\end{aligned}}} <p>and the aliquot sum s(12) of proper divisors is: </p> s ( 12 ) = 1 1 + 2 1 + 3 1 + 4 1 + 6 1 = 1 + 2 + 3 + 4 + 6 = 16. {\displaystyle {\begin{aligned}s(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}\\&=1+2+3+4+6=16.\end{aligned}}} <p><i>σ</i>−1(<i>n</i>) is sometimes called the <a href="/facts/Abundancy_index/K54Q9owk">abundancy index</a> of <i>n</i>, and we have: </p> σ − 1 ( 12 ) = 1 − 1 + 2 − 1 + 3 − 1 + 4 − 1 + 6 − 1 + 12 − 1 = 1 1 + 1 2 + 1 3 + 1 4 + 1 6 + 1 12 = 12 12 + 6 12 + 4 12 + 3 12 + 2 12 + 1 12 = 12 + 6 + 4 + 3 + 2 + 1 12 = 28 12 = 7 3 = σ 1 ( 12 ) 12 {\displaystyle {\begin{aligned}\sigma _{-1}(12)&=1^{-1}+2^{-1}+3^{-1}+4^{-1}+6^{-1}+12^{-1}\\[6pt]&={\tfrac {1}{1}}+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{12}}\\[6pt]&={\tfrac {12}{12}}+{\tfrac {6}{12}}+{\tfrac {4}{12}}+{\tfrac {3}{12}}+{\tfrac {2}{12}}+{\tfrac {1}{12}}\\[6pt]&={\tfrac {12+6+4+3+2+1}{12}}={\tfrac {28}{12}}={\tfrac {7}{3}}={\tfrac {\sigma _{1}(12)}{12}}\end{aligned}}} <h2 id="table-of-values">Table of values</h2> <p>The cases <i>x</i> = 2 to 5 are listed in <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A001157 through <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A001160, <i>x</i> = 6 to 24 are listed in <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013954 through <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A013972. </p> <table><tbody><tr><th><i>n</i></th><th>prime factorization</th><th>𝜎0(<i>n</i>)</th><th>𝜎1(<i>n</i>)</th><th>𝜎2(<i>n</i>)</th><th>𝜎3(<i>n</i>)</th><th>𝜎4(<i>n</i>)</th></tr><tr><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>2</td><td>2</td><td>2</td><td>3</td><td>5</td><td>9</td><td>17</td></tr><tr><td>3</td><td>3</td><td>2</td><td>4</td><td>10</td><td>28</td><td>82</td></tr><tr><td>4</td><td>22</td><td>3</td><td>7</td><td>21</td><td>73</td><td>273</td></tr><tr><td>5</td><td>5</td><td>2</td><td>6</td><td>26</td><td>126</td><td>626</td></tr><tr><td>6</td><td>2×3</td><td>4</td><td>12</td><td>50</td><td>252</td><td>1394</td></tr><tr><td>7</td><td>7</td><td>2</td><td>8</td><td>50</td><td>344</td><td>2402</td></tr><tr><td>8</td><td>23</td><td>4</td><td>15</td><td>85</td><td>585</td><td>4369</td></tr><tr><td>9</td><td>32</td><td>3</td><td>13</td><td>91</td><td>757</td><td>6643</td></tr><tr><td>10</td><td>2×5</td><td>4</td><td>18</td><td>130</td><td>1134</td><td>10642</td></tr><tr><td>11</td><td>11</td><td>2</td><td>12</td><td>122</td><td>1332</td><td>14642</td></tr><tr><td>12</td><td>22×3</td><td>6</td><td>28</td><td>210</td><td>2044</td><td>22386</td></tr><tr><td>13</td><td>13</td><td>2</td><td>14</td><td>170</td><td>2198</td><td>28562</td></tr><tr><td>14</td><td>2×7</td><td>4</td><td>24</td><td>250</td><td>3096</td><td>40834</td></tr><tr><td>15</td><td>3×5</td><td>4</td><td>24</td><td>260</td><td>3528</td><td>51332</td></tr><tr><td>16</td><td>24</td><td>5</td><td>31</td><td>341</td><td>4681</td><td>69905</td></tr><tr><td>17</td><td>17</td><td>2</td><td>18</td><td>290</td><td>4914</td><td>83522</td></tr><tr><td>18</td><td>2×32</td><td>6</td><td>39</td><td>455</td><td>6813</td><td>112931</td></tr><tr><td>19</td><td>19</td><td>2</td><td>20</td><td>362</td><td>6860</td><td>130322</td></tr><tr><td>20</td><td>22×5</td><td>6</td><td>42</td><td>546</td><td>9198</td><td>170898</td></tr><tr><td>21</td><td>3×7</td><td>4</td><td>32</td><td>500</td><td>9632</td><td>196964</td></tr><tr><td>22</td><td>2×11</td><td>4</td><td>36</td><td>610</td><td>11988</td><td>248914</td></tr><tr><td>23</td><td>23</td><td>2</td><td>24</td><td>530</td><td>12168</td><td>279842</td></tr><tr><td>24</td><td>23×3</td><td>8</td><td>60</td><td>850</td><td>16380</td><td>358258</td></tr><tr><td>25</td><td>52</td><td>3</td><td>31</td><td>651</td><td>15751</td><td>391251</td></tr><tr><td>26</td><td>2×13</td><td>4</td><td>42</td><td>850</td><td>19782</td><td>485554</td></tr><tr><td>27</td><td>33</td><td>4</td><td>40</td><td>820</td><td>20440</td><td>538084</td></tr><tr><td>28</td><td>22×7</td><td>6</td><td>56</td><td>1050</td><td>25112</td><td>655746</td></tr><tr><td>29</td><td>29</td><td>2</td><td>30</td><td>842</td><td>24390</td><td>707282</td></tr><tr><td>30</td><td>2×3×5</td><td>8</td><td>72</td><td>1300</td><td>31752</td><td>872644</td></tr><tr><td>31</td><td>31</td><td>2</td><td>32</td><td>962</td><td>29792</td><td>923522</td></tr><tr><td>32</td><td>25</td><td>6</td><td>63</td><td>1365</td><td>37449</td><td>1118481</td></tr><tr><td>33</td><td>3×11</td><td>4</td><td>48</td><td>1220</td><td>37296</td><td>1200644</td></tr><tr><td>34</td><td>2×17</td><td>4</td><td>54</td><td>1450</td><td>44226</td><td>1419874</td></tr><tr><td>35</td><td>5×7</td><td>4</td><td>48</td><td>1300</td><td>43344</td><td>1503652</td></tr><tr><td>36</td><td>22×32</td><td>9</td><td>91</td><td>1911</td><td>55261</td><td>1813539</td></tr><tr><td>37</td><td>37</td><td>2</td><td>38</td><td>1370</td><td>50654</td><td>1874162</td></tr><tr><td>38</td><td>2×19</td><td>4</td><td>60</td><td>1810</td><td>61740</td><td>2215474</td></tr><tr><td>39</td><td>3×13</td><td>4</td><td>56</td><td>1700</td><td>61544</td><td>2342084</td></tr><tr><td>40</td><td>23×5</td><td>8</td><td>90</td><td>2210</td><td>73710</td><td>2734994</td></tr><tr><td>41</td><td>41</td><td>2</td><td>42</td><td>1682</td><td>68922</td><td>2825762</td></tr><tr><td>42</td><td>2×3×7</td><td>8</td><td>96</td><td>2500</td><td>86688</td><td>3348388</td></tr><tr><td>43</td><td>43</td><td>2</td><td>44</td><td>1850</td><td>79508</td><td>3418802</td></tr><tr><td>44</td><td>22×11</td><td>6</td><td>84</td><td>2562</td><td>97236</td><td>3997266</td></tr><tr><td>45</td><td>32×5</td><td>6</td><td>78</td><td>2366</td><td>95382</td><td>4158518</td></tr><tr><td>46</td><td>2×23</td><td>4</td><td>72</td><td>2650</td><td>109512</td><td>4757314</td></tr><tr><td>47</td><td>47</td><td>2</td><td>48</td><td>2210</td><td>103824</td><td>4879682</td></tr><tr><td>48</td><td>24×3</td><td>10</td><td>124</td><td>3410</td><td>131068</td><td>5732210</td></tr><tr><td>49</td><td>72</td><td>3</td><td>57</td><td>2451</td><td>117993</td><td>5767203</td></tr><tr><td>50</td><td>2×52</td><td>6</td><td>93</td><td>3255</td><td>141759</td><td>6651267</td></tr></tbody></table> <h2 id="properties">Properties</h2> <h3>Formulas at prime powers</h3> <p>For a <a href="/facts/Prime_number/L20FLkgn">prime number</a> <i>p</i>, </p> σ 0 ( p ) = 2 σ 0 ( p n ) = n + 1 σ 1 ( p ) = p + 1 {\displaystyle {\begin{aligned}\sigma _{0}(p)&=2\\\sigma _{0}(p^{n})&=n+1\\\sigma _{1}(p)&=p+1\end{aligned}}} <p>because by definition, the factors of a prime number are 1 and itself. Also, where <i>pn</i># denotes the <a href="/facts/Primorial/NqlQBLDp">primorial</a>, </p> σ 0 ( p n # ) = 2 n {\displaystyle \sigma _{0}(p_{n}\#)=2^{n}} <p>since <i>n</i> prime factors allow a sequence of binary selection ( p i {\displaystyle p_{i}} or 1) from <i>n</i> terms for each proper divisor formed. However, these are not in general the smallest numbers whose number of divisors is a <a href="/facts/Power_of_two/MA6WEpTt">power of two</a>; instead, the smallest such number may be obtained by multiplying together the first <i>n</i> Fermi–Dirac primes, prime powers whose exponent is a power of two.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>Clearly, 1 < σ 0 ( n ) < n {\displaystyle 1<\sigma _{0}(n)<n} for all n > 2 {\displaystyle n>2} , and σ x ( n ) > n {\displaystyle \sigma _{x}(n)>n} for all n > 1 {\displaystyle n>1} , x > 0 {\displaystyle x>0} . </p><p>The divisor function is <a href="/facts/Multiplicative_function/4kfolbBl">multiplicative</a> (since each divisor <i>c</i> of the product <i>mn</i> with gcd ( m , n ) = 1 {\displaystyle \gcd(m,n)=1} distinctively correspond to a divisor <i>a</i> of <i>m</i> and a divisor <i>b</i> of <i>n</i>), but not <a href="/facts/Completely_multiplicative_function/4Sr9ADXQ">completely multiplicative</a>: </p> gcd ( a , b ) = 1 ⟹ σ x ( a b ) = σ x ( a ) σ x ( b ) . {\displaystyle \gcd(a,b)=1\Longrightarrow \sigma _{x}(ab)=\sigma _{x}(a)\sigma _{x}(b).} <p>The consequence of this is that, if we write </p> n = ∏ i = 1 r p i a i {\displaystyle n=\prod _{i=1}^{r}p_{i}^{a_{i}}} <p>where <i>r</i> = <i>ω</i>(<i>n</i>) is the <a href="/facts/Prime_omega_function/obfFNVEB">number of distinct prime factors</a> of <i>n</i>, <i>pi</i> is the <i>i</i>th prime factor, and <i>ai</i> is the maximum power of <i>pi</i> by which <i>n</i> is <a href="/facts/Divisible/DKNa2GvC">divisible</a>, then we have: <a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> σ x ( n ) = ∏ i = 1 r ∑ j = 0 a i p i j x = ∏ i = 1 r ( 1 + p i x + p i 2 x + ⋯ + p i a i x ) . {\displaystyle \sigma _{x}(n)=\prod _{i=1}^{r}\sum _{j=0}^{a_{i}}p_{i}^{jx}=\prod _{i=1}^{r}\left(1+p_{i}^{x}+p_{i}^{2x}+\cdots +p_{i}^{a_{i}x}\right).} <p>which, when <i>x</i> ≠ 0, is equivalent to the useful formula: <a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> σ x ( n ) = ∏ i = 1 r p i ( a i + 1 ) x − 1 p i x − 1 . {\displaystyle \sigma _{x}(n)=\prod _{i=1}^{r}{\frac {p_{i}^{(a_{i}+1)x}-1}{p_{i}^{x}-1}}.} <p>When <i>x</i> = 0, σ 0 ( n ) {\displaystyle \sigma _{0}(n)} is: <a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> σ 0 ( n ) = ∏ i = 1 r ( a i + 1 ) . {\displaystyle \sigma _{0}(n)=\prod _{i=1}^{r}(a_{i}+1).} <p>This result can be directly deduced from the fact that all divisors of n {\displaystyle n} are uniquely determined by the distinct tuples ( x 1 , x 2 , . . . , x i , . . . , x r ) {\displaystyle (x_{1},x_{2},...,x_{i},...,x_{r})} of integers with 0 ≤ x i ≤ a i {\displaystyle 0\leq x_{i}\leq a_{i}} (i.e. a i + 1 {\displaystyle a_{i}+1} independent choices for each x i {\displaystyle x_{i}} ). </p><p>For example, if <i>n</i> is 24, there are two prime factors (<i>p</i>1 is 2; <i>p</i>2 is 3); noting that 24 is the product of 23×31, <i>a</i>1 is 3 and <i>a</i>2 is 1. Thus we can calculate σ 0 ( 24 ) {\displaystyle \sigma _{0}(24)} as so: </p> σ 0 ( 24 ) = ∏ i = 1 2 ( a i + 1 ) = ( 3 + 1 ) ( 1 + 1 ) = 4 ⋅ 2 = 8. {\displaystyle \sigma _{0}(24)=\prod _{i=1}^{2}(a_{i}+1)=(3+1)(1+1)=4\cdot 2=8.} <p>The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24. </p> <h3>Other properties and identities</h3> <p><a href="/facts/Euler/z2fsbjgj">Euler</a> proved the remarkable recurrence:<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> σ 1 ( n ) = σ 1 ( n − 1 ) + σ 1 ( n − 2 ) − σ 1 ( n − 5 ) − σ 1 ( n − 7 ) + σ 1 ( n − 12 ) + σ 1 ( n − 15 ) + ⋯ = ∑ i ∈ N ( − 1 ) i + 1 ( σ 1 ( n − 1 2 ( 3 i 2 − i ) ) + σ 1 ( n − 1 2 ( 3 i 2 + i ) ) ) , {\displaystyle {\begin{aligned}\sigma _{1}(n)&=\sigma _{1}(n-1)+\sigma _{1}(n-2)-\sigma _{1}(n-5)-\sigma _{1}(n-7)+\sigma _{1}(n-12)+\sigma _{1}(n-15)+\cdots \\[12mu]&=\sum _{i\in \mathbb {N} }(-1)^{i+1}\left(\sigma _{1}\left(n-{\frac {1}{2}}\left(3i^{2}-i\right)\right)+\sigma _{1}\left(n-{\frac {1}{2}}\left(3i^{2}+i\right)\right)\right),\end{aligned}}} <p>where σ 1 ( 0 ) = n {\displaystyle \sigma _{1}(0)=n} if it occurs and σ 1 ( x ) = 0 {\displaystyle \sigma _{1}(x)=0} for x < 0 {\displaystyle x<0} , and 1 2 ( 3 i 2 ∓ i ) {\displaystyle {\tfrac {1}{2}}\left(3i^{2}\mp i\right)} are consecutive pairs of generalized <a href="/facts/Pentagonal_numbers/P8pNBERv">pentagonal numbers</a> (<a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A001318, starting at offset 1). Indeed, Euler proved this by logarithmic differentiation of the identity in his <a href="/facts/Pentagonal_number_theorem/z8TSDYAY">pentagonal number theorem</a>. </p><p>For a non-square integer, <i>n</i>, every divisor, <i>d</i>, of <i>n</i> is paired with divisor <i>n</i>/<i>d</i> of <i>n</i> and σ 0 ( n ) {\displaystyle \sigma _{0}(n)} is even; for a square integer, one divisor (namely n {\displaystyle {\sqrt {n}}} ) is not paired with a distinct divisor and σ 0 ( n ) {\displaystyle \sigma _{0}(n)} is odd. Similarly, the number σ 1 ( n ) {\displaystyle \sigma _{1}(n)} is odd if and only if <i>n</i> is a square or twice a square.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p><p>We also note <i>s</i>(<i>n</i>) = <i>σ</i>(<i>n</i>) − <i>n</i>. Here <i>s</i>(<i>n</i>) denotes the sum of the <i>proper</i> divisors of <i>n</i>, that is, the divisors of <i>n</i> excluding <i>n</i> itself. This function is used to recognize <a href="/facts/Perfect_number/WyqWtApV">perfect numbers</a>, which are the <i>n</i> such that <i>s</i>(<i>n</i>) = <i>n</i>. If <i>s</i>(<i>n</i>) > <i>n</i>, then <i>n</i> is an <a href="/facts/Abundant_number/K54Q9owk">abundant number</a>, and if <i>s</i>(<i>n</i>) < <i>n</i>, then <i>n</i> is a <a href="/facts/Deficient_number/dreFgv8A">deficient number</a>. </p><p>If n is a power of 2, n = 2 k {\displaystyle n=2^{k}} , then σ ( n ) = 2 ⋅ 2 k − 1 = 2 n − 1 {\displaystyle \sigma (n)=2\cdot 2^{k}-1=2n-1} and s ( n ) = n − 1 {\displaystyle s(n)=n-1} , which makes <i>n</i> <a href="/facts/Almost_perfect_number/7pzWByG0">almost-perfect</a>. </p><p>As an example, for two primes p , q : p < q {\displaystyle p,q:p<q} , let </p> n = p q {\displaystyle n=p\,q} . <p>Then </p> σ ( n ) = ( p + 1 ) ( q + 1 ) = n + 1 + ( p + q ) , {\displaystyle \sigma (n)=(p+1)(q+1)=n+1+(p+q),} φ ( n ) = ( p − 1 ) ( q − 1 ) = n + 1 − ( p + q ) , {\displaystyle \varphi (n)=(p-1)(q-1)=n+1-(p+q),} <p>and </p> n + 1 = ( σ ( n ) + φ ( n ) ) / 2 , {\displaystyle n+1=(\sigma (n)+\varphi (n))/2,} p + q = ( σ ( n ) − φ ( n ) ) / 2 , {\displaystyle p+q=(\sigma (n)-\varphi (n))/2,} <p>where φ ( n ) {\displaystyle \varphi (n)} is <a href="/facts/Euler_phi/mIQwXYzj">Euler's totient function</a>. </p><p>Then, the roots of </p> ( x − p ) ( x − q ) = x 2 − ( p + q ) x + n = x 2 − [ ( σ ( n ) − φ ( n ) ) / 2 ] x + [ ( σ ( n ) + φ ( n ) ) / 2 − 1 ] = 0 {\displaystyle (x-p)(x-q)=x^{2}-(p+q)x+n=x^{2}-[(\sigma (n)-\varphi (n))/2]x+[(\sigma (n)+\varphi (n))/2-1]=0} <p>express <i>p</i> and <i>q</i> in terms of <i>σ</i>(<i>n</i>) and <i>φ</i>(<i>n</i>) only, requiring no knowledge of <i>n</i> or p + q {\displaystyle p+q} , as </p> p = ( σ ( n ) − φ ( n ) ) / 4 − [ ( σ ( n ) − φ ( n ) ) / 4 ] 2 − [ ( σ ( n ) + φ ( n ) ) / 2 − 1 ] , {\displaystyle p=(\sigma (n)-\varphi (n))/4-{\sqrt {[(\sigma (n)-\varphi (n))/4]^{2}-[(\sigma (n)+\varphi (n))/2-1]}},} q = ( σ ( n ) − φ ( n ) ) / 4 + [ ( σ ( n ) − φ ( n ) ) / 4 ] 2 − [ ( σ ( n ) + φ ( n ) ) / 2 − 1 ] . {\displaystyle q=(\sigma (n)-\varphi (n))/4+{\sqrt {[(\sigma (n)-\varphi (n))/4]^{2}-[(\sigma (n)+\varphi (n))/2-1]}}.} <p>Also, knowing n and either σ ( n ) {\displaystyle \sigma (n)} or φ ( n ) {\displaystyle \varphi (n)} , or, alternatively, p + q {\displaystyle p+q} and either σ ( n ) {\displaystyle \sigma (n)} or φ ( n ) {\displaystyle \varphi (n)} allows an easy recovery of <i>p</i> and <i>q</i>. </p><p>In 1984, <a href="/facts/Roger_Heath-Brown/uMHxGtbd">Roger Heath-Brown</a> proved that the equality </p> σ 0 ( n ) = σ 0 ( n + 1 ) {\displaystyle \sigma _{0}(n)=\sigma _{0}(n+1)} <p>is true for infinitely many values of n, see <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A005237. </p> <h3>Dirichlet convolutions</h3> <p class="note">Main article: <a href="/facts/Dirichlet_convolution/DjuhDu1L">Dirichlet convolution</a></p> <p>By definition: σ = Id ∗ 1 {\displaystyle \sigma =\operatorname {Id} *\mathbf {1} } By <a href="/facts/M%C3%B6bius_inversion_formula/xDnSrPp6">Möbius inversion</a>: Id = σ ∗ μ {\displaystyle \operatorname {Id} =\sigma *\mu } </p> <h2 id="series-relations">Series relations</h2> <p>Two <a href="/facts/Dirichlet_series/uIAUaLcd">Dirichlet series</a> involving the divisor function are: <a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> ∑ n = 1 ∞ σ a ( n ) n s = ζ ( s ) ζ ( s − a ) for s > 1 , s > a + 1 , {\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)}{n^{s}}}=\zeta (s)\zeta (s-a)\quad {\text{for}}\quad s>1,s>a+1,} <p>where ζ {\displaystyle \zeta } is the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>. The series for <i>d</i>(<i>n</i>) = <i>σ</i>0(<i>n</i>) gives: <a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> ∑ n = 1 ∞ d ( n ) n s = ζ 2 ( s ) for s > 1 , {\displaystyle \sum _{n=1}^{\infty }{\frac {d(n)}{n^{s}}}=\zeta ^{2}(s)\quad {\text{for}}\quad s>1,} <p>and a <a href="/facts/Ramanujan/HBPr4bzd">Ramanujan</a> identity<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p> ∑ n = 1 ∞ σ a ( n ) σ b ( n ) n s = ζ ( s ) ζ ( s − a ) ζ ( s − b ) ζ ( s − a − b ) ζ ( 2 s − a − b ) , {\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)\sigma _{b}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-a)\zeta (s-b)\zeta (s-a-b)}{\zeta (2s-a-b)}},} <p>which is a special case of the <a href="/facts/Rankin%E2%80%93Selberg_method/naDkjNCF">Rankin–Selberg convolution</a>. </p><p>A <a href="/facts/Lambert_series/HFU5HfCR">Lambert series</a> involving the divisor function is: <a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> ∑ n = 1 ∞ q n σ a ( n ) = ∑ n = 1 ∞ ∑ j = 1 ∞ n a q j n = ∑ n = 1 ∞ n a q n 1 − q n = ∑ n = 1 ∞ Li − a ( q n ) {\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{a}(n)=\sum _{n=1}^{\infty }\sum _{j=1}^{\infty }n^{a}q^{j\,n}=\sum _{n=1}^{\infty }{\frac {n^{a}q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }\operatorname {Li} _{-a}(q^{n})} <p>for arbitrary <a href="/facts/Complex_number/2w7mqI7e">complex</a> |<i>q</i>| ≤ 1 and <i>a</i> ( Li {\displaystyle \operatorname {Li} } is the <a href="/facts/Polylogarithm/CKrkJMKn">polylogarithm</a>). This summation also appears as the <a href="/facts/Eisenstein_series/23vgJLFl">Fourier series of the Eisenstein series</a> and the <a href="/facts/Weierstrass_elliptic_functions/GHRXDknk">invariants of the Weierstrass elliptic functions</a>. </p><p>For k > 0 {\displaystyle k>0} , there is an explicit series representation with <a href="/facts/Ramanujan_sum/4xCStJrU">Ramanujan sums</a> c m ( n ) {\displaystyle c_{m}(n)} as :<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> σ k ( n ) = ζ ( k + 1 ) n k ∑ m = 1 ∞ c m ( n ) m k + 1 . {\displaystyle \sigma _{k}(n)=\zeta (k+1)n^{k}\sum _{m=1}^{\infty }{\frac {c_{m}(n)}{m^{k+1}}}.} <p>The computation of the first terms of c m ( n ) {\displaystyle c_{m}(n)} shows its oscillations around the "average value" ζ ( k + 1 ) n k {\displaystyle \zeta (k+1)n^{k}} : </p> σ k ( n ) = ζ ( k + 1 ) n k [ 1 + ( − 1 ) n 2 k + 1 + 2 cos 2 π n 3 3 k + 1 + 2 cos π n 2 4 k + 1 + ⋯ ] {\displaystyle \sigma _{k}(n)=\zeta (k+1)n^{k}\left[1+{\frac {(-1)^{n}}{2^{k+1}}}+{\frac {2\cos {\frac {2\pi n}{3}}}{3^{k+1}}}+{\frac {2\cos {\frac {\pi n}{2}}}{4^{k+1}}}+\cdots \right]} <h2 id="growth-rate">Growth rate</h2> <p>In <a href="/facts/Big_O_notation/weFFjSWg">little-o notation</a>, the divisor function satisfies the inequality:<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p> for all ε > 0 , d ( n ) = o ( n ε ) . {\displaystyle {\mbox{for all }}\varepsilon >0,\quad d(n)=o(n^{\varepsilon }).} <p>More precisely, <a href="/facts/Severin_Wigert/ECW9swg4">Severin Wigert</a> showed that:<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p> lim sup n → ∞ log d ( n ) log n / log log n = log 2. {\displaystyle \limsup _{n\to \infty }{\frac {\log d(n)}{\log n/\log \log n}}=\log 2.} <p>On the other hand, since <a href="/facts/Euclid%27s_theorem/Nl4SM0P0">there are infinitely many prime numbers</a>,<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> </p> lim inf n → ∞ d ( n ) = 2. {\displaystyle \liminf _{n\to \infty }d(n)=2.} <p>In <a href="/facts/Big-O_notation/weFFjSWg">Big-O notation</a>, <a href="/facts/Peter_Gustav_Lejeune_Dirichlet/b5UIU7Uy">Peter Gustav Lejeune Dirichlet</a> showed that the <a href="/facts/Average_order_of_an_arithmetic_function/yf0IfE84">average order</a> of the divisor function satisfies the following inequality:<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a><a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> </p> for all x ≥ 1 , ∑ n ≤ x d ( n ) = x log x + ( 2 γ − 1 ) x + O ( x ) , {\displaystyle {\mbox{for all }}x\geq 1,\sum _{n\leq x}d(n)=x\log x+(2\gamma -1)x+O({\sqrt {x}}),} <p>where γ {\displaystyle \gamma } is <a href="/facts/Euler%E2%80%93Mascheroni_constant/2kEgRvjC">Euler's gamma constant</a>. Improving the bound O ( x ) {\displaystyle O({\sqrt {x}})} in this formula is known as <a href="/facts/Divisor_summatory_function/LC52uv0F">Dirichlet's divisor problem</a>. </p><p>The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by: <a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> </p> lim sup n → ∞ σ ( n ) n log log n = e γ , {\displaystyle \limsup _{n\rightarrow \infty }{\frac {\sigma (n)}{n\,\log \log n}}=e^{\gamma },} <p>where lim sup is the <a href="/facts/Limit_superior/QeAhlygs">limit superior</a>. This result is <a href="/facts/Thomas_Hakon_Gr%C3%B6nwall/U9dqtxbJ">Grönwall</a>'s theorem, published in 1913 (Grönwall 1913). His proof uses <a href="/facts/Mertens%27_theorems/q92NHGaI">Mertens' third theorem</a>, which says that: </p> lim n → ∞ 1 log n ∏ p ≤ n p p − 1 = e γ , {\displaystyle \lim _{n\to \infty }{\frac {1}{\log n}}\prod _{p\leq n}{\frac {p}{p-1}}=e^{\gamma },} <p>where <i>p</i> denotes a prime. </p><p>In 1915, Ramanujan proved that under the assumption of the <a href="/facts/Riemann_hypothesis/Td8Jq5r0">Riemann hypothesis</a>, Robin's inequality </p> σ ( n ) < e γ n log log n {\displaystyle \ \sigma (n)<e^{\gamma }n\log \log n} (where γ is the <a href="/facts/Euler%E2%80%93Mascheroni_constant/2kEgRvjC">Euler–Mascheroni constant</a>) <p>holds for all sufficiently large <i>n</i> (Ramanujan 1997). The largest known value that violates the inequality is <i>n</i>=<a href="/facts/5040_(number)/BMhEgbL9">5040</a>. In 1984, Guy Robin proved that the inequality is true for all <i>n</i> > 5040 <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the Riemann hypothesis is true (Robin 1984). This is Robin's theorem and the inequality became known after him. Robin furthermore showed that if the Riemann hypothesis is false then there are an infinite number of values of <i>n</i> that violate the inequality, and it is known that the smallest such <i>n</i> > 5040 must be <a href="/facts/Superabundant_number/tJmXP8Ec">superabundant</a> (Akbary & Friggstad 2009). It has been shown that the inequality holds for large odd and square-free integers, and that the Riemann hypothesis is equivalent to the inequality just for <i>n</i> divisible by the fifth power of a prime (Choie et al. 2007). </p><p>Robin also proved, unconditionally, that the inequality: </p> σ ( n ) < e γ n log log n + 0.6483 n log log n {\displaystyle \ \sigma (n)<e^{\gamma }n\log \log n+{\frac {0.6483\ n}{\log \log n}}} <p>holds for all <i>n</i> ≥ 3. </p><p>A related bound was given by <a href="/facts/Jeffrey_Lagarias/HVvI3c3E">Jeffrey Lagarias</a> in 2002, who proved that the Riemann hypothesis is equivalent to the statement that: </p> σ ( n ) < H n + e H n log ( H n ) {\displaystyle \sigma (n)<H_{n}+e^{H_{n}}\log(H_{n})} <p>for every <a href="/facts/Natural_number/u65og8JE">natural number</a> <i>n</i> > 1, where H n {\displaystyle H_{n}} is the <i>n</i>th <a href="/facts/Harmonic_number/1b7v0hVn">harmonic number</a>, (Lagarias 2002). </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Arithmetic_function/hULxwsgm">Divisor sum convolutions</a>, lists a few identities involving the divisor functions</li> <li><a href="/facts/Euler%27s_totient_function/mIQwXYzj">Euler's totient function</a>, Euler's phi function</li> <li><a href="/facts/Refactorable_number/rgvsBS8G">Refactorable number</a></li> <li><a href="/facts/Table_of_divisors/3RXGmBtn">Table of divisors</a></li> <li><a href="/facts/Unitary_divisor/Nc6fpkaL">Unitary divisor</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Akbary, Amir; Friggstad, Zachary (2009), <a href="https://web.archive.org/web/20140411041855/http://webdocs.cs.ualberta.ca/~zacharyf/papers/superabundant.pdf">"Superabundant numbers and the Riemann hypothesis"</a> (PDF), <i>American Mathematical Monthly</i>, 116 (3): 273–275, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4169%2F193009709X470128">10.4169/193009709X470128</a>, archived from <a href="http://webdocs.cs.ualberta.ca/~zacharyf/Papers/superabundant.pdf">the original</a> (PDF) on 2014-04-11.</li> <li><a href="/facts/Tom_M._Apostol/S6vGtJWN">Apostol, Tom M.</a> (1976), <i>Introduction to analytic number theory</i>, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90163-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929">0434929</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0335.10001">0335.10001</a></li> <li><a href="/facts/Eric_Bach/QG7XPRWo">Bach, Eric</a>; <a href="/facts/Jeffrey_Shallit/4bmJgyIH">Shallit, Jeffrey</a>, <i>Algorithmic Number Theory</i>, volume 1, 1996, MIT Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-262-02405-5, see page 234 in section 8.8.</li> <li>Caveney, Geoffrey; <a href="/facts/Jean-Louis_Nicolas/3XJz1XLX">Nicolas, Jean-Louis</a>; Sondow, Jonathan (2011), <a href="http://www.integers-ejcnt.org/l33/l33.pdf">"Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis"</a> (PDF), <i>INTEGERS: The Electronic Journal of Combinatorial Number Theory</i>, 11: A33, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1110.5078">1110.5078</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2011arXiv1110.5078C">2011arXiv1110.5078C</a></li> <li><a href="/facts/YoungJu_Choie/ImNT1u09">Choie, YoungJu</a>; Lichiardopol, Nicolas; Moree, Pieter; Solé, Patrick (2007), "On Robin's criterion for the Riemann hypothesis", <i>Journal de théorie des nombres de Bordeaux</i>, 19 (2): 357–372, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.NT/0604314">math.NT/0604314</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.5802%2Fjtnb.591">10.5802/jtnb.591</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1246-7405">1246-7405</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2394891">2394891</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:3207238">3207238</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1163.11059">1163.11059</a></li> <li>Gioia, A. A.; Vaidya, A. M. (1967), "Amicable numbers with opposite parity", <i><a href="/facts/The_American_Mathematical_Monthly/Te6CHvkg">The American Mathematical Monthly</a></i>, 74 (8): 969–973, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2315280">10.2307/2315280</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2315280">2315280</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0220659">0220659</a></li> <li><a href="/facts/Thomas_Hakon_Gr%C3%B6nwall/U9dqtxbJ">Grönwall, Thomas Hakon</a> (1913), "Some asymptotic expressions in the theory of numbers", <i>Transactions of the American Mathematical Society</i>, 14: 113–122, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0002-9947-1913-1500940-6">10.1090/S0002-9947-1913-1500940-6</a></li> <li><a href="/facts/G._H._Hardy/zBkDLb05">Hardy, G. H.</a>; <a href="/facts/E._M._Wright/SS01ywc2">Wright, E. M.</a> (2008) [1938], <i>An Introduction to the Theory of Numbers</i>, Revised by <a href="/facts/Roger_Heath-Brown/uMHxGtbd">D. R. Heath-Brown</a> and <a href="/facts/Joseph_H._Silverman/acyPld7K">J. H. Silverman</a>. Foreword by <a href="/facts/Andrew_Wiles/bdc3k3pQ">Andrew Wiles</a>. (6th ed.), Oxford: <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-921986-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243">2445243</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1159.11001">1159.11001</a></li> <li>Ivić, Aleksandar (1985), <i>The Riemann zeta-function. 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(1972), <i>Elementary Introduction to Number Theory</i> (2nd ed.), Lexington: <a href="/facts/D._C._Heath_and_Company/qRMDoWMz">D. C. Heath and Company</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/77171950">77171950</a></li> <li>Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), <i>Elements of Number Theory</i>, Englewood Cliffs: <a href="/facts/Prentice_Hall/bynWAuat">Prentice Hall</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/77081766">77081766</a></li> <li><a href="/facts/Srinivasa_Ramanujan/HBPr4bzd">Ramanujan, Srinivasa</a> (1997), "Highly composite numbers, annotated by Jean-Louis Nicolas and Guy Robin", <i>The Ramanujan Journal</i>, 1 (2): 119–153, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1023%2FA%3A1009764017495">10.1023/A:1009764017495</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1382-4090">1382-4090</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1606180">1606180</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:115619659">115619659</a></li> <li>Robin, Guy (1984), "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", <i><a href="/facts/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9es/WSEmHp4v">Journal de Mathématiques Pures et Appliquées</a></i>, Neuvième Série, 63 (2): 187–213, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0021-7824">0021-7824</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0774171">0774171</a></li> <li>Williams, Kenneth S. (2011), <i>Number theory in the spirit of Liouville</i>, London Mathematical Society Student Texts, vol. 76, Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-17562-3, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1227.11002">1227.11002</a></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/DivisorFunction.html">"Divisor Function"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/RobinsTheorem.html">"Robin's Theorem"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="http://mathstat.carleton.ca/~williams/papers/pdf/249.pdf">Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions</a> PDF of a paper by Huard, Ou, Spearman, and Williams. Contains elementary (i.e. not relying on the theory of modular forms) proofs of divisor sum convolutions, formulas for the number of ways of representing a number as a sum of triangular numbers, and related results.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Long (1972, p. 46) - Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77171950 <a href="https://lccn.loc.gov/77171950" target="_blank">https://lccn.loc.gov/77171950</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Pettofrezzo & Byrkit (1970, p. 63) - Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77081766 <a href="https://lccn.loc.gov/77081766" target="_blank">https://lccn.loc.gov/77081766</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Long (1972, p. 46) - Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77171950 <a href="https://lccn.loc.gov/77171950" target="_blank">https://lccn.loc.gov/77171950</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Pettofrezzo & Byrkit (1970, p. 58) - Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77081766 <a href="https://lccn.loc.gov/77081766" target="_blank">https://lccn.loc.gov/77081766</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Ramanujan, S. (1915), "Highly Composite Numbers", Proceedings of the London Mathematical Society, s2-14 (1): 347–409, doi:10.1112/plms/s2_14.1.347; see section 47, pp. 405–406, reproduced in Collected Papers of Srinivasa Ramanujan, Cambridge Univ. Press, 2015, pp. 124–125 <a href="/wiki/Srinivasa_Ramanujan" target="_blank">/wiki/Srinivasa_Ramanujan</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Hardy & Wright (2008), pp. 310 f, §16.7. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Hardy & Wright (2008), pp. 310 f, §16.7. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Hardy & Wright (2008), pp. 310 f, §16.7. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Euler, Leonhard; Bell, Jordan (2004). "An observation on the sums of divisors". arXiv:math/0411587. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>https://scholarlycommons.pacific.edu/euler-works/175/, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs <a href="https://scholarlycommons.pacific.edu/euler-works/175/" target="_blank">https://scholarlycommons.pacific.edu/euler-works/175/</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>https://scholarlycommons.pacific.edu/euler-works/542/, De mirabilis proprietatibus numerorum pentagonalium <a href="https://scholarlycommons.pacific.edu/euler-works/542/" target="_blank">https://scholarlycommons.pacific.edu/euler-works/542/</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Gioia & Vaidya (1967). - Gioia, A. A.; Vaidya, A. M. (1967), "Amicable numbers with opposite parity", The American Mathematical Monthly, 74 (8): 969–973, doi:10.2307/2315280, JSTOR 2315280, MR 0220659 <a href="https://doi.org/10.2307%2F2315280" target="_blank">https://doi.org/10.2307%2F2315280</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Hardy & Wright (2008), pp. 326–328, §17.5. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Hardy & Wright (2008), pp. 326–328, §17.5. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Hardy & Wright (2008), pp. 334–337, §17.8. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Hardy & Wright (2008), pp. 338–341, §17.10. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>E. Krätzel (1981). Zahlentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften. p. 130. (German) <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Apostol (1976), p. 296. - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0434929</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Hardy & Wright (2008), pp. 342–347, §18.1. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Hardy & Wright (2008), pp. 342–347, §18.1. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Hardy & Wright (2008), pp. 342–347, §18.1. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Apostol (1976), Theorem 3.3. - Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0434929</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Hardy & Wright (2008), pp. 347–350, §18.2. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Hardy & Wright (2008), pp. 469–471, §22.9. - Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2445243</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> </ol>