A259981 - OEIS

%I #42 Jun 27 2017 15:36:24

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

%H Chai Wah Wu, <a href="/A259981/b259981.txt">Table of n, a(n) for n = 1..1000</a>

%H R. P. Boas & N. J. A. Sloane, <a href="/A005381/a005381.pdf">Correspondence, 1974</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AnomalousCancellation.html">Anomalous Cancellation</a>

%t 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 *)

%o (Python)

%o from sympy import primepi

%o def A002808(n):

%o ....m = n

%o ....while m != primepi(m) + 1 + n:

%o ........m += 1

%o ....return m

%o def A259981(n):

%o ....b, c = A002808(n), 0

%o ....for x in range(1,b):

%o ........for y in range(1,b):

%o ............if x != y:

%o ................w = b*(x-y)

%o ................for z in range(1,b):

%o ....................if x != z:

%o ........................if z*w == y*(x-z):

%o ............................c += 1

%o ....return c # _Chai Wah Wu_, Jul 15 2015

%Y Cf. A002808, A005381, A144925, A259983.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Jul 12 2015, following a suggestion from R. P. Boas, May 19 1974

%E Typo in a(61) corrected by _Chai Wah Wu_, Jul 15 2015