A292288 -id:A292288 - cp's OEIS frontend

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

Michael De Vlieger, Sep 15 2017

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

			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

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

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

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 *)

a(p) = 1.

a(p + 1) = p.

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

Original entry on oeis.org

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

Michael De Vlieger, Sep 13 2017

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.

			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

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

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

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 *)

a(p) = p^2 + p.

cp's OEIS Frontend

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

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