Divisor function: Difference between revisions

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where <math>\sigma(0)=n</math> if it occurs and <math>\sigma(i)=0</math> for <math>i \leq 0,</math> <math>\tfrac12 \left (3i^2-i \right )</math> are the [[pentagonal numbers]]. 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.{{Citation neededsfnp|date=May 2015Gioia|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'' for which ''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]].

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* {{Citation | last1=Caveney | first1=Geoffrey | last2=Nicolas | first2=Jean-Louis|author2-link=Jean-Louis Nicolas | last3=Sondow | first3=Jonathan | title=Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis | url=http://www.integers-ejcnt.org/l33/l33.pdf | year=2011 | journal=INTEGERS: The Electronic Journal of Combinatorial Number Theory | volume=11 | pages=A33| bibcode=2011arXiv1110.5078C | arxiv=1110.5078 }}

*{{Citation | last1=Choie | first1=YoungJu | author1-link= YoungJu Choie | last2=Lichiardopol | first2=Nicolas | last3=Moree | first3=Pieter | author3-link=Pieter Moree | last4=Solé | first4=Patrick | title=On Robin's criterion for the Riemann hypothesis | mr=2394891 | zbl=1163.11059 | arxiv=math.NT/0604314 | year=2007 | journal=Journal de théorie des nombres de Bordeaux | issn=1246-7405 | volume=19 | issue=2 | pages=357–372 | doi=10.5802/jtnb.591| s2cid=3207238 }}

*{{citation

| last1 = Gioia | first1 = A. A.

| last2 = Vaidya | first2 = A. M.

| doi = 10.2307/2315280

| journal = [[The American Mathematical Monthly]]

| jstor = 2315280

| mr = 220659

| pages = 969–973

| title = Amicable numbers with opposite parity

| volume = 74

| year = 1967}}

* {{Citation | last1=Grönwall | first1=Thomas Hakon | author1-link=Thomas Hakon Grönwall | title=Some asymptotic expressions in the theory of numbers | year=1913 | journal=Transactions of the American Mathematical Society | volume=14 | pages=113–122 | doi=10.1090/S0002-9947-1913-1500940-6| doi-access=free }}

* {{Citation | last1=Hardy | first1=G. H. | author1-link=G. H. Hardy | last2=Wright | first2=E. M. | author2-link=E. M. Wright | edition=6th | others=Revised by [[Roger Heath-Brown|D. R. Heath-Brown]] and [[Joseph H. Silverman|J. H. Silverman]]. Foreword by [[Andrew Wiles]]. | title=An Introduction to the Theory of Numbers | publisher=[[Oxford University Press]] | ___location=Oxford | isbn=978-0-19-921986-5 | mr=2445243 | zbl=1159.11001 | year=2008 | orig-year=1938}}