A292289 Smallest denominator of a proper fraction that has a nontrivial anomalous cancellation in base b.

6, 12, 14, 30, 33, 56, 60, 39, 64, 132, 138, 182, 189, 110, 84, 306, 315, 380, 390, 174, 272, 552, 564, 155, 402, 360, 259, 870, 885, 992, 1008, 405, 624, 609, 258, 1406, 1425, 754, 530, 1722, 1743, 1892, 1914, 504, 1120, 2256, 2280, 399, 1065, 1037, 897, 2862

Offset

2,1

Comments

See comments at A291093.

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

Smallest base b for which n/d, simplified, has a numerator greater than 1 is 51.

Links

Michael De Vlieger, Table of n, a(n) for n = 2..120

Michael De Vlieger, Base-b proper fractions n/d having nontrivial anomalous cancellation, with 2 <= b <= 120 and d <= b^2 + b.

Eric Weisstein's World of Mathematics, Anomalous Cancellation

Formula

a(p) = p^2 + p.

Example

a(5) = 30, the corresponding numerator is 6; these are written "11/110" in quinary, cancelling a 1 in both numerator and denominator yields "1/10" which is 1/5. 6/30 = 1/5.
Table of smallest values correlated with least numerators:
b = base and index.
n = smallest numerator that pertains to d.
d = smallest denominator that has a nontrivial anomalous cancellation in base b (this sequence).
n/d = simplified ratio of numerator n and denominator d.
k = base-b digit cancelled in the numerator and denominator to arrive at n/d.
b-n+1 = difference between base and numerator plus one.
b^2-d = difference between the square of the base and denominator.
.
   b     n      d   n/d     k  b-n+1  b^2-d
   -----------------------------------------
   2     3      6   1/2     1    0     -2
   3     4     12   1/3     1    0     -3
   4     7     14   1/2     3    2      2
   5     6     30   1/5     1    0     -5
   6    11     33   1/3     5    4      3
   7     8     56   1/7     1    0     -7
   8    15     60   1/4     7    6      4
   9    13     39   1/3     4    3     42
  10    16     64   1/4     6    5     36
  11    12    132   1/11    1    0    -11
  12    23    138   1/6    11   10      6
  13    14    182   1/13    1    0    -13
  14    27    189   1/7    13   12      7
  15    22    110   1/5     7    6    115
  16    21     84   1/4     5    4    172

Mathematica

Table[SelectFirst[Range[b, b^2 + b], Function[m, 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}]] ] != {}]], {b, 2, 30}] (* Michael De Vlieger, Sep 13 2017 *)

Crossrefs

Cf. A291093/A291094, A292288 (numerators), A292393 (digit that is canceled).

Sequence in context: A259397 A183029 A113791 * A281352 A351843 A135763

Adjacent sequences: A292286 A292287 A292288 * A292290 A292291 A292292

Keyword

nonn,frac,base

Author

Michael De Vlieger, Sep 13 2017

Status

approved