Abundance
المؤلف:
Guy, R. K.
المصدر:
Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, 2004.
الجزء والصفحة:
...
21-11-2020
857
Abundance
The abundance of a number 
, sometimes also called the abundancy (a term which in this work, is reserved for a different but related quantity), is the quantity
where 
is the divisor function. The abundances of 
, 2, ... are 
, 
, 
, 
, 
, 0, 
, 
, 
, 
, 
, 4, 
, 
, 
, 
, ... (OEIS A033880).
The following table lists special classifications given to a number 
based on the value of 
.
 |
class |
OEIS |
list of  |
 |
deficient number |
A005100 |
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, ... |
 |
almost perfect number |
A000079 |
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, ... |
| 0 |
perfect number |
A000396 |
6, 28, 496, 8128, ... |
| 1 |
quasiperfect number |
|
none known |
 |
abundant number |
A005101 |
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, ... |
Values of 
such that 
is odd are given by 
, 2, 4, 8, 9, 16, 18, 25, 32, ... (OEIS A028982; i.e., the union of nonzero squares and twice the squares). Values of 
such that 
is square are given by 
, 12, 28, 70, 88, 108, 168, ... (OEIS A109510).
REFERENCES:
Guy, R. K. Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, 2004.
Sloane, N. J. A. Sequences A000079/M1129, A000396/M4186, A005100/M0514, A005101/M4825, A028982, A033880, and A109510 in "The On-Line Encyclopedia of Integer Sequences."
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