Erdős–Nicolas number

{{Short description|In math, a number that is equal to the sum of some of its factors}}

{{Infobox integer sequence

| named_after = Paul Erdős, Jean-Louis Nicolas

| publication_year = 1975

| author = Erdős, P., Nicolas, J. L.

| parentsequence = {{nowrap|Abundant numbers}}

| first_terms = 24, 2016, 8190

| largest_known_term = 3304572752464376776401640967110656

| OEIS = A194472

| OEIS_name = Erdős-Nicolas numbers

}}

In number theory, an Erdős–Nicolas number is a number that is not perfect, but that equals one of the partial sums of its divisors.

That is, a number {{mvar|n}} is an Erdős–Nicolas number when there exists another number {{mvar|m}} such that

: $\backslash sum\_\{d\backslash mid\; n,\backslash \; d\backslash leq\; m\}d=n.$

{{cite book

|last = De Koninck

|first = Jean-Marie

|year = 2009

|title = Those Fascinating Numbers

|publisher = American Mathematical Soc.

|page = 141

|isbn = 978-0-8218-4807-4

|url = https://www.ams.org/bookpages/mbk-64

}}

The first ten Erdős–Nicolas numbers are

:24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, 61900800 and 91963648. ({{oeis|A194472}})

They are named after Paul Erdős and Jean-Louis Nicolas, who wrote about them in 1975.{{citation

|first1 = P. |last1 = Erdős | author1-link=Paul Erdős

|first2 = J.L. |last2 = Nicolas | author2-link=Jean-Louis Nicolas

|title = Répartition des nombres superabondants

|journal = Bull. Soc. Math. France

|issue = 103

|year = 1975

|volume = 79 |pages = 65–90

|doi = 10.24033/bsmf.1793 |url = http://archive.numdam.org/article/BSMF_1975__103__65_0.pdf

|zbl=0306.10025

}}