Granville number

In mathematics, specifically number theory, Granville numbers, also known as $\mathcal$-perfect numbers, are an extension of the perfect numbers.

The Granville set

In 1996, Andrew Granville proposed the following construction of a set $\mathcal$:

:Let $1\in\mathcal$, and for any integer $n$ larger than 1, let $n\in$ if
::$\sum_ d \leq n.$

A Granville number is an element of $\mathcal$ 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 $\mathcal$. Granville numbers are also called $\mathcal$-perfect numbers.

General properties

The elements of $\mathcal$ can be -deficient, -perfect, or -abundant. In particular, 2-perfect numbers are a proper subset of $\mathcal$.

S-deficient numbers

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

::$\sum_d <$

S-perfect numbers

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

:6, 24, 28, 96, 126, 224, 384, 496, 1536, 1792, 6144, 8128, 14336, ...

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

S-abundant numbers

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

::$\sum_d >$

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

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

Examples

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

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

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

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

References

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Number theory