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A number n ≥ 10080 {\displaystyle n\geq 10080} is called extremely abundant number, if either n = 10080 {\displaystyle n=10080} or n > 10080 {\displaystyle n>10080} and for all 10080 ≤ m < n {\displaystyle 10080\leq m<n}
σ ( m ) m log log m < σ ( n ) n log log n {\displaystyle {\frac {\sigma (m)}{m\log \log m}}<{\frac {\sigma (n)}{n\log \log n}}}
First 10 extremely abundant numbers:
1) ( 7 ♯ ) ( 3 ♯ ) 2 3 = 10080 {\displaystyle \ (7\sharp )(3\sharp )2^{3}=10080}
2) ( 113 ♯ ) ( 13 ♯ ) ( 5 ♯ ) ( 3 ♯ ) 2 2 3 {\displaystyle \ (113\sharp )(13\sharp )(5\sharp )(3\sharp )^{2}2^{3}}
3) ( 127 ♯ ) ( 13 ♯ ) ( 5 ♯ ) ( 3 ♯ ) 2 2 3 {\displaystyle \ (127\sharp )(13\sharp )(5\sharp )(3\sharp )^{2}2^{3}}
4) ( 131 ♯ ) ( 13 ♯ ) ( 5 ♯ ) ( 3 ♯ ) 2 2 3 {\displaystyle \ (131\sharp )(13\sharp )(5\sharp )(3\sharp )^{2}2^{3}}
5) ( 137 ♯ ) ( 13 ♯ ) ( 5 ♯ ) ( 3 ♯ ) 2 2 3 {\displaystyle \ (137\sharp )(13\sharp )(5\sharp )(3\sharp )^{2}2^{3}}
6) ( 139 ♯ ) ( 13 ♯ ) ( 5 ♯ ) ( 3 ♯ ) 2 2 3 {\displaystyle \ (139\sharp )(13\sharp )(5\sharp )(3\sharp )^{2}2^{3}}
7) ( 139 ♯ ) ( 13 ♯ ) ( 5 ♯ ) ( 3 ♯ ) 2 2 4 {\displaystyle \ (139\sharp )(13\sharp )(5\sharp )(3\sharp )^{2}2^{4}}
8) ( 151 ♯ ) ( 13 ♯ ) ( 5 ♯ ) ( 3 ♯ ) 2 2 3 {\displaystyle \ (151\sharp )(13\sharp )(5\sharp )(3\sharp )^{2}2^{3}}
9) ( 151 ♯ ) ( 13 ♯ ) ( 5 ♯ ) ( 3 ♯ ) 2 2 4 {\displaystyle \ (151\sharp )(13\sharp )(5\sharp )(3\sharp )^{2}2^{4}}
10) ( 151 ♯ ) ( 13 ♯ ) ( 7 ♯ ) ( 3 ♯ ) 2 2 4 {\displaystyle \ (151\sharp )(13\sharp )(7\sharp )(3\sharp )^{2}2^{4}}
where
p k ♯ = ∏ j = 1 k p j {\displaystyle p_{k}\sharp =\prod _{j=1}^{k}p_{j}}