In mathematics, specifically number theory, Granville numbers, also known as S {\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 S {\displaystyle {\mathcal {S}}} :
Let 1 ∈ S {\displaystyle 1\in {\mathcal {S}}} , and for any integer n {\displaystyle n} larger than 1, let n ∈ S {\displaystyle n\in {\mathcal {S}}} if
∑ d ∣ n , d < n , d ∈ S d ≤ n . {\displaystyle \sum _{d\mid n,\;d<n,\;d\in {\mathcal {S}}}d\leq n.}
A Granville number is an element of S {\displaystyle {\mathcal {S}}} for which equality holds, that is, n {\displaystyle n} is a Granville number if it is equal to the sum of its proper divisors that are also in S {\displaystyle {\mathcal {S}}} . Granville numbers are also called S {\displaystyle {\mathcal {S}}} -perfect numbers.
General properties The elements of S {\displaystyle {\mathcal {S}}} can be k-deficient, k-perfect, or k-abundant. In particular, 2-perfect numbers are a proper subset of S {\displaystyle {\mathcal {S}}} .
S-deficient numbers Numbers that fulfill the strict form of the inequality in the above definition are known as S {\displaystyle {\mathcal {S}}} -deficient numbers. That is, the S {\displaystyle {\mathcal {S}}} -deficient numbers are the natural numbers for which the sum of their divisors in S {\displaystyle {\mathcal {S}}} is strictly less than themselves:
∑ d ∣ n , d < n , d ∈ S d < n {\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 S {\displaystyle {\mathcal {S}}} -perfect numbers. That is, the S {\displaystyle {\mathcal {S}}} -perfect numbers are the natural numbers that are equal the sum of their divisors in S {\displaystyle {\mathcal {S}}} . The first few S {\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 S {\displaystyle {\mathcal {S}}} -perfect. However, there are numbers such as 24 which are S {\displaystyle {\mathcal {S}}} -perfect but not perfect. The only known S {\displaystyle {\mathcal {S}}} -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 S {\displaystyle {\mathcal {S}}} -abundant numbers. That is, the S {\displaystyle {\mathcal {S}}} -abundant numbers are the natural numbers for which the sum of their divisors in S {\displaystyle {\mathcal {S}}} is strictly greater than themselves:
∑ d ∣ n , d < n , d ∈ S d > n {\displaystyle \sum _{d\mid {n},\;d<n,\;d\in {\mathcal {S}}}d>{n}}
They belong to the complement of S {\displaystyle {\mathcal {S}}} . The first few S {\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)
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