A291094 Denominators of fractions with nontrivial anomalous cancellation, listed with multiplicity if multiple numerators are possible.

64, 65, 95, 98, 110, 120, 121, 130, 132, 136, 140, 143, 150, 154, 160, 160, 165, 170, 176, 180, 187, 190, 190, 192, 194, 195, 196, 196, 198, 202, 204, 206, 208, 210, 220, 220, 230, 231, 231, 238, 238, 240, 242, 242, 250

Offset: 1

Author

N. J. A. Sloane, Aug 21 2017

Keywords

• nonn
• frac
• base

Comments

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.

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.

From Jon E. Schoenfield, Sep 12 2017: (Start)

For k = 1..12, the smallest denominator D that appears exactly k times and its corresponding numerators are as follows:

.

k D numerators

== ==== ================================================

1 64 16

2 160 16 64

3 294 49 98 196

4 392 49 98 196 294

5 490 49 98 196 294 392

6 660 66 165 264 363 462 561

7 770 77 176 275 374 473 572 671

8 880 88 187 286 385 484 583 682 781

9 990 99 198 297 396 495 594 693 792 891

10 1980 99 198 297 396 495 594 693 792 891 990

11 2970 99 198 297 396 495 594 693 792 891 990 1980

12 3960 99 198 297 396 495 594 693 792 891 990 1980 2970

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).

(End)

Examples

			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.
		

References

• 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.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..8544 (denominators d <= 10^4; first 169 terms from N. J. A. Sloane)
• Michael De Vlieger, Correlation of A291093 and A291094 and their ratio (for denominators d <= 10^4)
• B. L. Schwartz, Proposal 434, Mathematics Magazine Vol. 34, No. 3 (1961), Problems and Questions, p. 173.
• N. J. A. Sloane, Maple program.
• N. J. A. Sloane, List of first 169 fractions (file gives line number, numerator, denominator).
• Eric W. Weisstein, Anomalous Cancellation.

Crossrefs

See A291093 for numerators.

Cf. A159975/A159976, A290462/A290463.

Cf. A291965/A291966 for a variant.

Cf. A367206, A367207.

Programs

Mathematica

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 *)