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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
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
KEYWORD
nonn,frac,base
AUTHOR
Michael De Vlieger, Sep 15 2017
STATUS
approved

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Last modified April 20 07:43 EDT 2024. Contains 371799 sequences. (Running on oeis4.)