A292288 Numerators of smallest denominator of a proper fraction that has a nontrivial anomalous cancellation in base b.

3, 4, 7, 6, 11, 8, 15, 13, 16, 12, 23, 14, 27, 22, 21, 18, 35, 20, 39, 29, 34, 24, 47, 31, 67, 40, 37, 30, 59, 32, 63, 45, 52, 87, 43, 38, 75, 58, 53, 42, 83, 44, 87, 56, 70, 48, 95, 57, 71, 122, 69, 54, 107, 67, 71, 77, 88, 60, 119, 62, 123, 94, 73, 81, 79

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 bases b for which n/d, simplified, has a numerator greater than 1 are 51, 77, 92, ...

See link "Base-b proper fractions ..." below for more information. - Michael De Vlieger, Sep 18 2017

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 + 1), for prime p.

Example

a(5) = 6, the corresponding denominator is 30; 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 (this sequence).
d = smallest denominator that has a nontrivial anomalous cancellation in base b.
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[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}][[All, 1]] (* Michael De Vlieger, Sep 15 2017 *)

Crossrefs

Cf. A291093/A291094, A292289 (denominators), A292393 (digit that is canceled).

Sequence in context: A057032 A347529 A279388 * A347273 A355584 A113957

Adjacent sequences: A292285 A292286 A292287 * A292289 A292290 A292291

Keyword

nonn,frac,base

Author

Michael De Vlieger, Sep 15 2017

Status

approved