1,1

The set of all proper solutions up to 3-digit denominators is given by 13/325, 16/64, 19/95, 26/65, 124/217, 127/762, 138/184, 139/973, 145/435, 148/185, 154/253, 161/644, 163/326, 166/664, 176/275, 182/819, 187/286, 187/385, 187/748, 199/995, 218/981, 266/665, 273/728, 275/374, 286/385, 316/632, 327/872, 364/637, 412/721, and 436/763. The concept of anomalous cancellation can be extended to arbitrary bases. Prime bases have no solutions, but there is a solution corresponding to each proper divisor of a composite b. When b-1 is prime, this type of solution is the only one. For base 4, for example, the only solution is 32(base 4)/13(base 4) = 2(base 4). Boas gives a table of solutions for b<40. The number of solutions is even unless b is an even square.

Boas, R. P. "Anomalous Cancellation." Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113-129, 1979.

Moessner, A. Scripta Math. 19; 20.

Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 86-87, 1988.

Eric W. Weisstein, Anomalous Cancellation

A159975(n)/a(n) is a proper fraction which undergoes Anomalous Cancellation.

The first four values are the only four such cases for numerator and denominators of two digits in base 10: a(1) = 325 because 13/325 if you cancel a digit "3" in numerator and denominator correctly yields the simplification 1/25. a(2) = 64 because 16/64 if you cancel a digit "6" in numerator and denominator correctly yields the simplification 1/4. a(3) = 95 because 19/95 if you cancel a digit "9" in numerator and denominator correctly yields the simplification 1/5. a(4) = 65 because 26/65 if you cancel a digit "6" in numerator and denominator correctly yields the simplification 2/5.

Cf. A159975.

Sequence in context: A132644 A013763 A013887 this_sequence A159844 A000443 A097101

Adjacent sequences: A159973 A159974 A159975 this_sequence A159977 A159978 A159979

base,fini,frac,full,nonn,uned

Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 28 2009