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A291093 Form the list of fractions with nontrivial anomalous cancellation, sorted first by denominators, then by numerators; sequence lists the numerators.

%I A291093 #55 Feb 16 2025 08:33:50
%S A291093 16,26,19,49,11,12,22,13,33,34,14,44,15,55,16,64,66,17,77,18,88,19,95,
%T A291093 96,97,39,49,98,99,101,102,103,104,21,22,121,23,33,132,34,136,24,44,
%U A291093 143,25,55,154,26,65,66,165,106,67,27,77,176,28,88,187,29
%N A291093 Form the list of fractions with nontrivial anomalous cancellation, sorted first by denominators, then by numerators; sequence lists the numerators.
%C A291093 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.
%C A291093 Nontrivial means that fractions of the form x0/y0 are excluded (otherwise there would be a large number of trivial entries like 120/340).
%C A291093 The fractions are assumed to be in the range 0 to 1, and of course are not reduced.
%C A291093 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.
%C A291093 A fraction is listed only once, even if the cancellation is possible in more than one way.
%D A291093 R. P. Boas, "Anomalous Cancellation." Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113-129, 1979.
%D A291093 A. Moessner, Scripta Math. 19; 20.
%D A291093 C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory. New York: Dover, 1988, pp. 86-87.
%H A291093 Michael De Vlieger, <a href="/A291093/b291093.txt">Table of n, a(n) for n = 1..8544</a> (numerators for denominators d <= 10^4; first 169 terms from N. J. A. Sloane)
%H A291093 Michael De Vlieger, <a href="/A291093/a291093_2.txt">Correlation of A291093 and A291094 and their ratio</a> (all denominators d <= 10^4)
%H A291093 B. L. Schwartz, <a href="http://doi.org/10.2307/2688504">Proposal 434</a>, Mathematics Magazine Vol. 34, No. 3 (1961), Problems and Questions, p. 173.
%H A291093 N. J. A. Sloane, <a href="/A291093/a291093.txt">Maple program</a>.
%H A291093 N. J. A. Sloane, <a href="/A291093/a291093_1.txt">List of first 169 fractions</a> (file gives line number, numerator, denominator).
%H A291093 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/AnomalousCancellation.html">Anomalous Cancellation</a>.
%e A291093 The first two terms correspond to the fractions 16/64 = 1/4 (cancel the 6!) and 26/65 = 2/5 (again cancel the 6!).
%e A291093 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.
%t A291093 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 *)
%Y A291093 See A291094 for denominators.
%Y A291093 Cf. A159975/A159976, A290462/A290463, A291966/A291965.
%K A291093 nonn,frac,base
%O A291093 1,1
%A A291093 _N. J. A. Sloane_, Aug 21 2017