A094907 Number of different nontrivial two-digit cancellations of the form (xy)/(zx) = y/z in base n.

0, 0, 1, 0, 2, 0, 2, 2, 4, 0, 4, 0, 2, 6, 7, 0, 4, 0, 4, 10, 6, 0, 6, 6, 4, 6, 10, 0, 6, 0, 4, 8, 6, 6, 21, 0, 2, 6, 18, 0, 6, 0, 4, 18, 10, 0, 8, 10, 10, 12, 12, 0, 6, 16, 22, 14, 6, 0, 10, 0, 2, 12, 21, 12, 20, 0, 4, 10, 22, 0, 10, 0, 2, 12, 20, 14, 24, 0, 8, 24, 8, 0, 10, 28, 6, 6, 18, 0, 10

Offset: 2

Author

Rick Mabry (rmabry(AT)pilot.lsus.edu), Jun 16 2004

Keywords

• nonn
• base

Comments

Trivial cancellations are of the form xx/xx=x/x, e.g. 44/44 = 4/4.

Examples

			a(10) = 4 because we have the four nontrivial base-10 cancellations 64/16 = 4/1, 65/26 = 5/2, 95/19 = 5/1, 98/49 = 8/4.
		

References

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

Links

• Ludovic Schwob, Table of n, a(n) for n = 2..1000
• R. P. Boas, Anomalous Cancellation, The Two Year College Mathematics Journal, Vol. 3, No. 2 (Autumn 1972), 21-24.
• Eric Weisstein's World of Mathematics, Anomalous Cancellation.

Programs

Mathematica

a[n_]:= Length[(DeleteCases[ #1, {u_, u_, u_}] & )[ Position[Table[(n*x + y)/(n*z + x) == y/z, {x, 1, n - 1}, {y, 1, x - 1}, {z, 1, y - 1}], True]]]

Formula

From Ludovic Schwob, Nov 10 2020: (Start)

a(n)=0 if and only if n is prime.

a(n) is odd if and only if n is an even square. (End)