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A291093
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Form the list of fractions with nontrivial anomalous cancellation, sorted first by denominators, then by numerators; sequence lists the numerators.
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9
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16, 26, 19, 49, 11, 12, 22, 13, 33, 34, 14, 44, 15, 55, 16, 64, 66, 17, 77, 18, 88, 19, 95, 96, 97, 39, 49, 98, 99, 101, 102, 103, 104, 21, 22, 121, 23, 33, 132, 34, 136, 24, 44, 143, 25, 55, 154, 26, 65, 66, 165, 106, 67, 27, 77, 176, 28, 88, 187, 29
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OFFSET
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1,1
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COMMENTS
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An unreduced fraction N/D is said to have the anomalous cancellation property if there is a single digit that can be cancelled from both N and D without changing the value of the fraction. The first and most famous example is 16/64 = 1/4 after cancelling the 6's.
Nontrivial means that fractions of the form x0/y0 are excluded (otherwise there would be a large number of trivial entries like 120/340).
The fractions are assumed to be in the range 0 to 1, and of course are not reduced.
The denominators d are considered in the order 11, 12, 13, ..., and then the numerators are considered in the order n = 10, 11, 12, ..., d-1.
A fraction is listed only once, even if the cancellation is possible in more than one way.
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REFERENCES
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R. P. Boas, "Anomalous Cancellation." Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113-129, 1979.
A. Moessner, Scripta Math. 19; 20.
C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory. New York: Dover, 1988, pp. 86-87.
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LINKS
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B. L. Schwartz, Proposal 434, Mathematics Magazine Vol. 34, No. 3 (1961), Problems and Questions, p. 173.
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EXAMPLE
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The first two terms correspond to the fractions 16/64 = 1/4 (cancel the 6!) and 26/65 = 2/5 (again cancel the 6!).
The first 20 fractions are (before cancellation) 16/64, 26/65, 19/95, 49/98, 11/110, 12/120, 22/121, 13/130, 33/132, 34/136, 14/140, 44/143, 15/150, 55/154, 16/160, 64/160, 66/165, 17/170, 77/176, 18/180, which equal (after cancellation) 1/4, 2/5, 1/5, 1/2, 1/10, 1/10, 2/11, 1/10, 1/4, 1/4, 1/10, 4/13, 1/10, 5/14, 1/10, 2/5, 2/5, 1/10, 7/16, 1/10.
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MATHEMATICA
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Flatten@ Table[Select[Range[11, m - 1], Function[k, Function[{r, w, n, d}, AnyTrue[Flatten@ Map[Apply[Outer[Divide, #1, #2] &, #] &, Transpose@ MapAt[# /. 0 -> Nothing &, Map[Function[x, Map[Map[FromDigits@ Delete[x, #] &, Position[x, #]] &, Intersection @@ {n, d}]], {n, d}], -1]], # == Divide @@ {k, m} &]] @@ {k/m, #, First@ #, Last@ #} &@ Map[IntegerDigits, {k, m}] - Boole[Mod[{k, m}, 10] == {0, 0}]]], {m, 290}] (* Michael De Vlieger, Sep 13 2017 *)
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CROSSREFS
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KEYWORD
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nonn,frac,base
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AUTHOR
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STATUS
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approved
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