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%I #46 Nov 12 2023 12:52:40
%S 64,65,95,98,110,120,121,130,132,136,140,143,150,154,160,160,165,170,
%T 176,180,187,190,190,192,194,195,196,196,198,202,204,206,208,210,220,
%U 220,230,231,231,238,238,240,242,242,250
%N Denominators of fractions with nontrivial anomalous cancellation, listed with multiplicity if multiple numerators are possible.
%C An unreduced fraction N/D is said to have the anomalous cancellation property if there is a single digit that can be canceled from both N and D without changing the value of the fraction. The first and most famous example is 16/64 = 1/4 after canceling the 6's.
%C 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 The fractions are assumed to be in the range 0 to 1, and of course are not reduced.
%C 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 A fraction is listed only once, even if the cancellation is possible in more than one way.
%C From _Jon E. Schoenfield_, Sep 12 2017: (Start)
%C For k = 1..12, the smallest denominator D that appears exactly k times and its corresponding numerators are as follows:
%C .
%C k D numerators
%C == ==== ================================================
%C 1 64 16
%C 2 160 16 64
%C 3 294 49 98 196
%C 4 392 49 98 196 294
%C 5 490 49 98 196 294 392
%C 6 660 66 165 264 363 462 561
%C 7 770 77 176 275 374 473 572 671
%C 8 880 88 187 286 385 484 583 682 781
%C 9 990 99 198 297 396 495 594 693 792 891
%C 10 1980 99 198 297 396 495 594 693 792 891 990
%C 11 2970 99 198 297 396 495 594 693 792 891 990 1980
%C 12 3960 99 198 297 396 495 594 693 792 891 990 1980 2970
%C Smallest denominator that appears exactly k times in the sequence for k = 1..41: 64, 160, 294, 392, 490, 660, 770, 880, 990, 1980, 2970, 3960, 4950, 5830, 6710, 7920, 8910, 9900, 11940, 12935, 13065, 14925, 15920, 16080, 16915, 18905, 19095, 23952, 24950, 25948, 26052, 24309, 28942, 29940, 29058, 31396, 32934, 34068, 33932, 35928, 36926 (note that this sequence is nonmonotonic; e.g., its 29th and 32nd terms are 24950 and 24309, respectively).
%C (End)
%D R. P. Boas, "Anomalous Cancellation." Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113-129, 1979.
%D A. Moessner, Scripta Math. 19; 20.
%D C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory. New York: Dover, 1988, pp. 86-87.
%H Michael De Vlieger, <a href="/A291094/b291094.txt">Table of n, a(n) for n = 1..8544</a> (denominators d <= 10^4; first 169 terms from N. J. A. Sloane)
%H Michael De Vlieger, <a href="/A291094/a291094.txt">Correlation of A291093 and A291094 and their ratio</a> (for denominators d <= 10^4)
%H 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 N. J. A. Sloane, <a href="/A291093/a291093.txt">Maple program</a>.
%H N. J. A. Sloane, <a href="/A291093/a291093_1.txt">List of first 169 fractions</a> (file gives line number, numerator, denominator).
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/AnomalousCancellation.html">Anomalous Cancellation</a>.
%e 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 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 Flatten@ Table[ConstantArray[m, Count[Range[11, m - 1], _?(Function[k, Function[{r, 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, 250}] (* _Michael De Vlieger_, Sep 13 2017 *)
%Y See A291093 for numerators.
%Y Cf. A159975/A159976, A290462/A290463.
%Y Cf. A291965/A291966 for a variant.
%Y Cf. A367206, A367207.
%K nonn,frac,base
%O 1,1
%A _N. J. A. Sloane_, Aug 21 2017
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