Hemiperfect number

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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):

k Smallest k-hemiperfect number Number of digits
3 2 1
5 24 2
7 4320 4
9 8910720 7
11 17116004505600 14
13 170974031122008628879954060917200710847692800 45
15 12749472205565550032020636281352368036406720997031277595140988449695952806020854579200000[1] 89
17 27172904004644864174776390325441204588387876949911859015099963347683477337589882757168182488651338324482275518065870009252589097916253652597707421065171952334010184222064839170719744000000000[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. ^ a b "Number Theory". Numericana.com. Retrieved 2012-08-21. 


Retrieved from "https://en.wikipedia.org/w/index.php?title=Hemiperfect_number&oldid=821228753"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Hemiperfect_number
This page is based on the copyrighted Wikipedia article "Hemiperfect number"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA