A259981 Let b be the n-th composite number, A002808(n); a(n) is number of base-b digits x,y,z such that (xb+y)/(zb+x)=y/z.

1, 2, 2, 2, 4, 4, 2, 6, 7, 4, 4, 10, 6, 6, 6, 4, 6, 10, 6, 4, 8, 6, 6, 21, 2, 6, 18, 6, 4, 18, 10, 8, 10, 10, 12, 12, 6, 16, 22, 14, 6, 10, 2, 12, 21, 12, 20, 4, 10, 22, 10, 2, 12, 20, 14, 24, 8, 24, 8, 10, 28, 6, 6, 18, 10, 28, 16, 10, 6, 6, 30, 4, 24, 37, 6, 6, 46, 14, 10, 6, 18, 24, 6, 18

Offset: 1

Author

N. J. A. Sloane, Jul 12 2015, following a suggestion from R. P. Boas, May 19 1974

Keywords

• nonn

Comments

R. P. Boas writes (slightly edited): The problem originated in the rather silly observation that 64/16 = 4/1 ("cancel" the 6's). I once asked what happens in base b, i.e., when is (xb+y)/(zb+x) = y/z? There are no nontrivial instances of the cancellation phenomenon when b is prime, so we restrict b to A002808; the sequence gives the number of instances of the phenomenon for each composite b. When b-1 is prime the only instances have x=b-1 and the number of them is the number of proper divisors of b (see A144925). A259983 is the subsequence of this sequence corresponding to bases b in A005381.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1000
• R. P. Boas & N. J. A. Sloane, Correspondence, 1974
• Eric Weisstein's World of Mathematics, Anomalous Cancellation

Crossrefs

Cf. A002808, A005381, A144925, A259983.

Programs

Mathematica

Table[Count[Flatten[Table[(x b + y) z == y (z b + x), {x, b}, {y, b}, {z, y - 1}], 2], True], {b, Select[Range[115], CompositeQ]}] (* Eric W. Weisstein, Oct 16 2015 *)

Python

from sympy import primepi
def A002808(n):
    m = n
    while m != primepi(m) + 1 + n:
        m += 1
    return m
def A259981(n):
    b, c = A002808(n), 0
    for x in range(1,b):
        for y in range(1,b):
            if x != y:
                w = b*(x-y)
                for z in range(1,b):
                    if x != z:
                        if z*w == y*(x-z):
                            c += 1
    return c # Chai Wah Wu, Jul 15 2015

Extensions

Typo in a(61) corrected by Chai Wah Wu, Jul 15 2015