A292393 Base-n digit k involved in anomalous cancellation in the proper fraction A292288(n)/A292289(n).

1, 1, 3, 1, 5, 1, 7, 4, 6, 1, 11, 1, 13, 7, 5, 1, 17, 1, 19, 8, 12, 1, 23, 6, 15, 13, 9, 1, 29, 1, 31, 12, 18, 17, 7, 1, 37, 19, 13, 1, 41, 1, 43, 11, 24, 1, 47, 8, 21, 20, 17, 1, 53, 12, 15, 20, 30, 1, 59, 1, 61, 31, 9, 16, 13, 1, 67, 24, 23, 1, 71, 1, 73, 37

Offset: 2

Author

Michael De Vlieger, Sep 15 2017

Keywords

• nonn
• base

Comments

For prime base p, (p + 1)/(p^2 + p) simplifies to 1/p by canceling digit k = 1 in the numerator and denominator. This fraction is written "11/110" in base p and simplifies to "1/10" = 1/p.

See link "Base-b proper fractions n/d having nontrivial anomalous cancellation, with 2 <= b <= 120 and d <= b^2 + b" at A292289 for more information. - Michael De Vlieger, Sep 18 2017

Examples

			a(10) = 6, since A292288(10)/A292289(10) = 16/64 = 1/4; we can "cancel" k = 6 in the numerator and the denominator and obtain 1/4 anomalously.
a(12) = 11, since A292288(12)/A292289(12) = 23/138 = "1b/b6" in base 12, where "b" represents digit 11. This fraction simplifies to 1/6. Digit "b" = 11 is canceled and "anomalously" yields 1/6.
a(16) = 5, since A292288(16)/A292289(16) = 21/84 = hexadecimal "15/54". This fraction simplifies to 1/4. We can "cancel" k = 5 in the numerator and denominator and obtain 1/4 anomalously.
Table relating a(n) with A292288(n) and A292289(n).
n = base and index.
N = A292288(n) = smallest numerator that pertains to D.
D = A292289(n) = smallest denominator that has a nontrivial anomalous cancellation in base n.
n/d = simplified ratio of numerator N and denominator D.
k = a(n) = base-n digit anomalously canceled in the numerator and denominator to arrive at N/D.
.
   n     N       D   N/D       k
  ------------------------------
   2     3       6   1/2       1
   3     4      12   1/3       1
   4     7      14   1/2       3
   5     6      30   1/5       1
   6    11      33   1/3       5
   7     8      56   1/7       1
   8    15      60   1/4       7
   9    13      39   1/3       4
  10    16      64   1/4       6
  11    12     132   1/11      1
  12    23     138   1/6      11
  13    14     182   1/13      1
  14    27     189   1/7      13
  15    22     110   1/5       7
  16    21      84   1/4       5
  17    18     306   1/17      1
  18    35     315   1/9      17
  19    20     380   1/19      1
  20    39     390   1/10     19
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..120
• Eric W. Weisstein, Anomalous Cancellation

Crossrefs

Cf. A292288 (numerators), A292289 (denominators).

Programs

Mathematica

Table[Intersection[IntegerDigits[#1, b], IntegerDigits[#2, b]] & @@ Flatten@ Catch@ Do[If[Length@ # > 0, Throw[#], #] &@ Map[{#, m} &, #] &@ Select[Range[b + 1, 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[#, b] &@ Delete[x, #] &, Position[x, #]] &, Intersection @@ {n, d}]], {n, d}], -1]], # == Divide @@ {k, m} &]] @@ {k/m, #, First@ #, Last@ #} &@ Map[IntegerDigits[#, b] &, {k, m}] - Boole[Mod[{k, m}, b] == {0, 0}]] ], {m, b, b^2 + b}], {b, 2, 30}] // Flatten (* Michael De Vlieger, Sep 15 2017 *)

Formula

a(p) = 1.

a(p + 1) = p.