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A292288 Numerators of smallest denominator of a proper fraction that has a nontrivial anomalous cancellation in base b. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A126253 A057032 A279388 * A113957 A073185 A284341

Adjacent sequences:  A292285 A292286 A292287 * A292289 A292290 A292291

KEYWORD

nonn,frac,base

AUTHOR

Michael De Vlieger, Sep 15 2017

STATUS

approved

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Last modified April 14 15:18 EDT 2021. Contains 342949 sequences. (Running on oeis4.)