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

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

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 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.

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 (numerators for denominators d <= 10^4; first 169 terms from N. J. A. Sloane)
• Michael De Vlieger, Correlation of A291093 and A291094 and their ratio (all 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 A291094 for denominators.

Cf. A159975/A159976, A290462/A290463, A291966/A291965.

Programs

Mathematica

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 *)