Granville number

Last updated September 24, 2023

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

The Granville set

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

Let

1\in {\mathcal {S}}

, and for any integer

n

larger than 1, let

n\in {\mathcal {S}}

if

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

A Granville number is an element of ${\mathcal {S}}$ 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 ${\mathcal {S}}$ . Granville numbers are also called ${\mathcal {S}}$ -perfect numbers.^[2]

General properties

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

S-deficient numbers

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

\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 ${\mathcal {S}}$ -perfect numbers.^[1] That is, the ${\mathcal {S}}$ -perfect numbers are the natural numbers that are equal the sum of their divisors in ${\mathcal {S}}$ . The first few ${\mathcal {S}}$ -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 ${\mathcal {S}}$ -perfect.^[1] However, there are numbers such as 24 which are ${\mathcal {S}}$ -perfect but not perfect. The only known ${\mathcal {S}}$ -perfect number with three distinct prime factors is 126 = 2 · 3² · 7.^[2]

S-abundant numbers

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

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

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

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

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

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

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References

1 2 3 4 5 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.
1 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.

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[De_Koninck_(1998)-1] 1 2 3 4 5 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.

[De_Koninck_2009-2] 1 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.

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