Divisor function: Difference between revisions

Revision as of 01:49, 26 February 2022

Revision as of 03:19, 26 February 2022

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===Other properties and identities===

[[Euler]] proved the remarkable recurrence:<ref>{{Cite arXiv |eprint = math/0411587|last1 = Euler|first1 = Leonhard|title = An observation on the sums of divisors|last2 = Bell|first2 = Jordan|year = 2004}}</ref><ref>httphttps://eulerarchivescholarlycommons.maapacific.orgedu/euler-works/pages175/E175.html, ''DecouverteDécouverte d'une loi tout extraordinaire des nombres par rapport aà la somme de leurs diviseurs''</ref><ref>https://scholarlycommons.pacific.edu/euler-works/542/, ''De mirabilis proprietatibus numerorum pentagonalium''</ref>

:<math>\begin{align}

\sigma(n) &= \sigma(n-1)+\sigma(n-2)-\sigma(n-5)-\sigma(n-7)+\sigma(n-12)+\sigma(n-15)+ \cdots \\[12mu]

&= \sum_{i\in\Z} (-1)^{i+1}\left( \sigma \left ( n-\frac12frac{1}{2} \left ( 3i^2-i \right ) \right) )+ \sigma \left (n n-\frac12frac{1}{2} \left ( 3i^2+i \right ) \right ) \right)

\end{align}</math>

where <math>\sigma(0)=n</math> if it occurs and <math>\sigma(i)=0</math> for <math>i \leq 0,</math>, and <math>\tfrac12tfrac{1}{2} \left ( 3i^2- \mp i \right )</math> are theconsecutive pairs of generalized [[pentagonal numbers]] ({{OEIS2C|A001318}}, starting at offset 1). Indeed, Euler proved this by logarithmic differentiation of the identity in his [[Pentagonal number theorem]].

For a non-square integer, ''n'', every divisor, ''d'', of ''n'' is paired with divisor ''n''/''d'' of ''n'' and <math>\sigma_{0}(n)</math> is even; for a square integer, one divisor (namely <math>\sqrt n</math>) is not paired with a distinct divisor and <math>\sigma_{0}(n)</math> is odd. Similarly, the number <math>\sigma_{1}(n)</math> is odd if and only if ''n'' is a square or twice a square.{{sfnp|Gioia|Vaidya|1967}}

We also note ''s''(''n'') = ''σ''(''n'')&nbsp;−&nbsp;''n''. Here ''s''(''n'') denotes the sum of the ''proper'' divisors of ''n'', that is, the divisors of ''n'' excluding ''n'' itself. This function is the one used to recognize [[perfect number]]s, which are the ''n'' forsuch whichthat ''s''(''n'') =&nbsp;''n''. If ''s''(''n'') > ''n'', then ''n'' is an [[abundant number]], and if ''s''(''n'') < ''n'', then ''n'' is a [[deficient number]].

If {{mvar|n}} is a power of 2, for example, <math>n = 2^k</math>, then <math>\sigma(n) = 2 \cdot 2^k - 1 = 2n - 1,</math> and ''<math>s(n) = n - 1''</math>, which makes ''n'' [[Almost perfect number|almost-perfect]].

As an example, for two distinct primes ''<math>p'' and '',q'' with '':p < q''</math>, let

:<math>n = pq. p\,q</math>.

Then

:<math>\sigma(n) = (p+1)(q+1) = n + 1 + (p+q), </math>

:<math>\varphi(n) = (p-1)(q-1) = n + 1 - (p+q), </math>

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where <math>\varphi(n)</math> is [[Euler phi|Euler's totient function]].

Then, the roots of:

:<math>(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 </math>

allow us to express ''p'' and ''q'' in terms of ''σ''(''n'') and ''φ''(''n'') only, withoutrequiring evenno knowingknowledge of ''n'' or ''<math>p+q''</math>, as:

:<math>p = (\sigma(n) - \varphi(n))/4 - \sqrt{[(\sigma(n) - \varphi(n))/4]^2 - [(\sigma(n) + \varphi(n))/2 - 1]}, </math>

:<math>q = (\sigma(n) - \varphi(n))/4 + \sqrt{[(\sigma(n) - \varphi(n))/4]^2 - [(\sigma(n) + \varphi(n))/2 - 1]}. </math>

Also, knowing {{mvar|n}} and either <math>\sigma(n)</math> or <math>\varphi(n)</math>, (or, knowingalternatively, <math>p+q</math> and either <math>\sigma(n)</math> or <math>\varphi(n)</math>) allows usan toeasy easilyrecovery findof ''p'' and ''q''.

In 1984, [[Roger Heath-Brown]] proved that the equality

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:<math>\sigma_0(n) = \sigma_0(n + 1)</math>

is true for aninfinitely infinity ofmany values of {{mvar|n}}, see {{OEIS2C|A005237}}.

==Series relations==