High abundant number

For all natural number m which the high abundant number (こうどかじょうすう British: highly abundant number) is natural number, and are m <n

${\displaystyle \sigma (m)<\sigma (n)}$

を is natural number n to satisfy. But σ is a divisor function. Specifically

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108,... (progression A002093 of the online integer row Dictionary)

である. I use the name called the abundant number, but all altitude abundant number may not be abundant number. Deficient number, 6 are whole number, and seven altitude abundant number 1 of the beginning in particular, 2, 3, 4, 8, 10 are not abundant number. All the high abundant number more than 12 is abundant number.

The high abundant number was defined by Pillai and developed by Alaoglu and Erdős. Alaoglu and Erdős expressed high abundant number to 10⁴.

For example, 5 is not high abundant number. This is because it becomes σ(5) = 5+1 = 6, and it is σ(4) = 4 +2 +1 = 7, and 4 smaller than 5 is bigger than σ(5). 8 is high abundant number. This is because it becomes σ(8) = 8 +4 +2 +1 = 15, and the number more than σ(8) does not exist with a number less than 8.

The odd altitude abundant number is only 1 and 3. ^[1]

Table of contents

Allied with other numbers

• The factorial from 1 to 8 is high abundant number, but all factorial is not high abundant number. For example,

σ (9!) =σ(362880) = 1481040,

However, the sum of divisor bigger than the sum of this divisor exists with a number smaller than this number.

σ(360360) = 1572480,

Thus, 9 is not は altitude abundant number.

• Alaoglu and Erdős showed that all super abundant number was high abundant number. And I expected that there would be the altitude abundant number that was not super abundant number in infinity. This was shown to be right by Jean-Louis Nicolas.

• 7200 is biggest 多冪数 with index part of all prime factors more than 2 in the high abundant number. (7200 = 2⁵*32*⁵2) all altitude abundant number bigger than this has one independent prime factor (the prime factor that an index part is not more than 2 in). Therefore, 7200 is the biggest altitude abundant number that the total of the divisor becomes the odd number. ^[2]

Footnote

1. ^ See Alaoglu & Erdős (1944)
2. ^ Alaoglu & Erdős (1944), pp. 464–466.

References

• Alaoglu, L.; Erdős, P. (1944). "On highly composite and similar numbers". Transactions of the American Mathematical Society 56 (3): 448–469. doi: 10.2307/1990319. JSTOR 1990319. MR 0011087. http://combinatorica.hu/~p_erdos/1944-03.pdf. 
• Nicolas, Jean-Louis (1969). "Ordre maximal d'unélément du groupe Sn des permutations et "highly composite numbers"". Bull. Soc. Math. It is 129–191. France 97 It is http://www.numdam.org/item?id=BSMF_1969__97__129_0. MR 0254130 
• Pillai, S. S. (1943). "Highly abundant numbers." Bull. Calcutta Math. Soc. 35: 141–156. MR 0010560. 

Allied item

• Abundant number
• Super abundant number

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