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Date: 2-11-2020
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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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