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

Author

Michael De Vlieger, Sep 13 2017

Keywords

• nonn
• frac
• base

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.

Examples

			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
		

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

Crossrefs

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

Programs

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 *)

Formula

a(p) = p^2 + p.