Quoting mathworld:
Anomalous cancellation is a "canceling" of digits of $a$ and $b$ in the numerator and denominator of a fraction $a/b$ which results in a fraction equal to the original. Note that if there are multiple but differing counts of one or more digits in the numerator and denominator there is ambiguity about which digits to cancel, so it is simplest to exclude such cases from consideration.
Some examples, here are all the 2 digit fractions that cancel:
$$ \require{cancel} \frac{16}{64} = \frac{1 \cancel{6}}{\cancel{6}4} = \frac{1}{4} \\ \frac{19}{95} = \frac{1}{5} \\ \frac{26}{65} = \frac{2}{5} \\ \frac{49}{98} = \frac{4}{8} $$
I wrote a simple program to iterate over values and check for anomalous cancellation. Mathworld says
The numbers of anomalously cancelling proper fractions having $n$ digits in both numerator and denominator for $n$=1, 2, ... are 0, 4, 161, 1851, ....
However, for $n$=3, I'm coming up with 157 results. My first few values are:
Cancel 6: 106 / 265 == 10 / 25
Cancel 6: 116 / 464 == 11 / 44
Cancel 9: 119 / 595 == 11 / 55
Cancel 3: 132 / 231 == 12 / 21
Cancel 3: 134 / 536 == 14 / 56
Cancel 3: 134 / 737 == 14 / 77
Cancel 3: 134 / 938 == 14 / 98
Cancel 3: 136 / 238 == 16 / 28
I am not considering a factor of $0$, or allowing a factor $x \cong 0 \mod 10$ or allowing any numerator or denominator divisible by 10. I am not considering a single factor that occurs more than once. I am considering any proper substring for my factors, for example, a four digit number would consider any one, two, and three digit factors that appear as substrings. (Edit: this was not considering disjoint combinations of factors, nor was it considering two or more digit combination of factors; see update below).
Likewise, for $n$ or fewer digits, mathworld gives a count of 190 while I'm coming up with 180.
It seems the definition from mathworld is incomplete, so my question is:
What definition should be used that will result in the counts given on the mathworld page?
update
Mathworld lists $\frac{1016}{4064} = \frac{101}{404}$ so it appears that uncancelled digits ($0$ in this case) are allowed.
Mathworld also lists $\frac{1019}{5095} = \frac{11}{55}$ so it appears that disjoint combinations are allowed.
$\frac{499}{998} = \frac{49}{98} = \frac{4}{8}$ Should this be counted once or twice? Note: there are 4 such three digit fractions.
Mathworld says that if there are multiple but differing counts of a digit the fraction should be ignored. That seems reasonable, but is perhaps more restrictive than it needs to be because there are unambiguous counter examples such as $\frac{796}{995} = \frac{76}{95}$
I'm pulling out combinations of consecutive digits of a number to consider as factors. For example, given $123$, I will consider $\{3\}, \{2\}, \{23\}, \{1\}, \{1,3\}, \{12\}$. I'm still not allowing any factor $x \cong 0 \mod 10$. Additionally, because this is supposed to be anomalous, I am not allowing any factor that divides the numerator or denominator outright. This gets me 134 unique three digit fractions, 27 short of 161. Allowing factors that do divide the numerator or denominator gives 183 unique three digit fractions.
