Hemiperfect number

In number theory, a hemiperfect number is a positive integer with a half-integral abundancy index.

For a given odd number k, a number n is called k-hemiperfect if and only if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to k/2 × n.

Smallest k-hemiperfect numbers

The following table gives an overview of the smallest k-hemiperfect numbers for k ≤ 17 (sequence A088912 in the OEIS):

kSmallest k-hemiperfect number Number of digits
32 1
524 2
74,320 4
98,910,720 7
1117,116,004,505,600 14
13170,974,031,122,008,628,879,954,060,917,200,710,847,692,800 45
1512,749,472,205,565,550,032,020,636,281,352,368,036,406,720,997,031,277,595,140,988,449,695,952,806,020,854,579,200,000[1] 89
1727,172,904,004,644,864,174,776,390,325,441,204,588,387,876,949,911,859,015,099,963,347,683,477,337,589,882,757,168,182,488,651,338,324,482,275,518,065,870,009,252,589,097,916,253,652,597,707,421,065,171,952,334,010,184,222,064,839,170,719,744,000,000,000[1] 191

For example, 24 is 5-hemiperfect because the sum of the divisors of 24 is

1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = 5/2 × 24.

See also

References
  1. "Number Theory". Numericana.com. Retrieved 2012-08-21.


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