Granville number

In 1996, Andrew Granville proposed the following construction of a set $\backslash mathcal\{S\}$:{{cite journal|vauthors=De Koninck JM, Ivić A|title=On a Sum of Divisors Problem|journal=Publications de l'Institut mathématique|year=1996|volume=64|issue=78|pages=9–20|url=http://www.emis.de/journals/PIMB/078/n078p009.pdf|access-date=27 March 2011}}

:Let $1\backslash in\backslash mathcal\{S\}$, and for any integer $n$ larger than 1, let $n\backslash in\{\backslash mathcal\{S\}\}$ if

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A Granville number is an element of $\backslash mathcal\{S\}$ for which equality holds, that is, $n$ is a Granville number if it is equal to the sum of its proper divisors that are also in $\backslash mathcal\{S\}$. Granville numbers are also called $\backslash mathcal\{S\}$-perfect numbers.{{Cite book |last=de Koninck |first=Jean-Marie |author-link=Jean-Marie De Koninck |translator-first = J. M. |translator-last=de Koninck |title=Those Fascinating Numbers |url=https://archive.org/details/thosefascinating0000koni/page/40/mode/2up |url-access=registration |publisher=American Mathematical Society |location= Providence, RI |year=2008 |page=40 |isbn=978-0-8218-4807-4 |oclc=317778112 |mr=2532459 }}

The elements of $\backslash mathcal\{S\}$ can be {{math|k}}-deficient, {{math|k}}-perfect, or {{math|k}}-abundant. In particular, 2-perfect numbers are a proper subset of $\backslash mathcal\{S\}$.

Numbers that fulfill the strict form of the inequality in the above definition are known as $\backslash mathcal\{S\}$-deficient numbers. That is, the $\backslash mathcal\{S\}$-deficient numbers are the natural numbers for which the sum of their divisors in $\backslash mathcal\{S\}$ is strictly less than themselves:

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Numbers that fulfill equality in the above definition are known as $\backslash mathcal\{S\}$-perfect numbers. That is, the $\backslash mathcal\{S\}$-perfect numbers are the natural numbers that are equal the sum of their divisors in $\backslash mathcal\{S\}$. The first few $\backslash mathcal\{S\}$-perfect numbers are:

:6, 24, 28, 96, 126, 224, 384, 496, 1536, 1792, 6144, 8128, 14336, ... {{OEIS|A118372}}

Every perfect number is also $\backslash mathcal\{S\}$-perfect. However, there are numbers such as 24 which are $\backslash mathcal\{S\}$-perfect but not perfect. The only known $\backslash mathcal\{S\}$-perfect number with three distinct prime factors is 126 = 2 · 3² · 7.

Every number of form 2^(n - 1) * (2^n - 1) * (2^n)^m where m >= 0, where 2^n - 1 is Prime, are Granville Numbers. So, there are infinitely many Granville Numbers and the infinite family has 2 prime factors- 2 and a Mersenne Prime. Others include 126, 5540590, 9078520, 22528935, 56918394 and 246650552 having 3, 5, 5, 5, 5 and 5 prime factors.

Numbers that violate the inequality in the above definition are known as $\backslash mathcal\{S\}$-abundant numbers. That is, the $\backslash mathcal\{S\}$-abundant numbers are the natural numbers for which the sum of their divisors in $\backslash mathcal\{S\}$ is strictly greater than themselves:

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They belong to the complement of $\backslash mathcal\{S\}$. The first few $\backslash mathcal\{S\}$-abundant numbers are:

:12, 18, 20, 30, 42, 48, 56, 66, 70, 72, 78, 80, 84, 88, 90, 102, 104, ... {{OEIS|A181487}}

Every deficient number and every perfect number is in $\backslash mathcal\{S\}$ because the restriction of the divisors sum to members of $\backslash mathcal\{S\}$ either decreases the divisors sum or leaves it unchanged. The first natural number that is not in $\backslash mathcal\{S\}$ is the smallest abundant number, which is 12. The next two abundant numbers, 18 and 20, are also not in $\backslash mathcal\{S\}$. However, the fourth abundant number, 24, is in $\backslash mathcal\{S\}$ because the sum of its proper divisors in $\backslash mathcal\{S\}$ is:

:1 + 2 + 3 + 4 + 6 + 8 = 24

In other words, 24 is abundant but not $\backslash mathcal\{S\}$-abundant because 12 is not in $\backslash mathcal\{S\}$. In fact, 24 is $\backslash mathcal\{S\}$-perfect - it is the smallest number that is $\backslash mathcal\{S\}$-perfect but not perfect.

The smallest odd abundant number that is in $\backslash mathcal\{S\}$ is 2835, and the smallest pair of consecutive numbers that are not in $\backslash mathcal\{S\}$ are 5984 and 5985.

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Category:Number theory