Granville number

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

The Granville set

In 1996, Andrew Granville proposed the following construction of a set ${\displaystyle {\mathcal {S}}}$: ^[1]

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

${\displaystyle \sum _{d\mid n,\;d<n,\;d\in {\mathcal {S}}}d\leq n.}$

A Granville number is an element of ${\displaystyle {\mathcal {S}}}$ for which equality holds, that is, ${\displaystyle n}$ is a Granville number if it is equal to the sum of its proper divisors that are also in ${\displaystyle {\mathcal {S}}}$. Granville numbers are also called ${\displaystyle {\mathcal {S}}}$-perfect numbers. ^[2]

General properties

The elements of ${\displaystyle {\mathcal {S}}}$ can be k-deficient, k-perfect, or k-abundant. In particular, 2-perfect numbers are a proper subset of ${\displaystyle {\mathcal {S}}}$. ^[1]

S-deficient numbers

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

${\displaystyle \sum _{d\mid {n},\;d<n,\;d\in {\mathcal {S}}}d<{n}}$

S-perfect numbers

Numbers that fulfill equality in the above definition are known as ${\displaystyle {\mathcal {S}}}$-perfect numbers. ^[1] That is, the ${\displaystyle {\mathcal {S}}}$-perfect numbers are the natural numbers that are equal the sum of their divisors in ${\displaystyle {\mathcal {S}}}$. The first few ${\displaystyle {\mathcal {S}}}$-perfect numbers are:

6, 24, 28, 96, 126, 224, 384, 496, 1536, 1792, 6144, 8128, 14336, ... (sequence A118372 in the OEIS )

Every perfect number is also ${\displaystyle {\mathcal {S}}}$-perfect. ^[1] However, there are numbers such as 24 which are ${\displaystyle {\mathcal {S}}}$-perfect but not perfect. The only known ${\displaystyle {\mathcal {S}}}$-perfect number with three distinct prime factors is 126 = 2 · 3² · 7. ^[2]

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.

S-abundant numbers

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

${\displaystyle \sum _{d\mid {n},\;d<n,\;d\in {\mathcal {S}}}d>{n}}$

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

12, 18, 20, 30, 42, 48, 56, 66, 70, 72, 78, 80, 84, 88, 90, 102, 104, ... (sequence A181487 in the OEIS )

Examples

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

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

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

The smallest odd abundant number that is in ${\displaystyle {\mathcal {S}}}$ is 2835, and the smallest pair of consecutive numbers that are not in ${\displaystyle {\mathcal {S}}}$ are 5984 and 5985. ^[1]

References

1. De Koninck JM, Ivić A (1996). "On a Sum of Divisors Problem" (PDF). Publications de l'Institut mathématique. 64 (78): 9–20. Retrieved 27 March 2011.
2. de Koninck, Jean-Marie (2008). Those Fascinating Numbers . Translated by de Koninck, J. M. Providence, RI: American Mathematical Society. p. 40. ISBN   978-0-8218-4807-4. MR   2532459. OCLC   317778112.