This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A292393 #10 Feb 16 2025 08:33:51
%S A292393 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,
%T A292393 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,
%U A292393 20,30,1,59,1,61,31,9,16,13,1,67,24,23,1,71,1,73,37
%N A292393 Base-n digit k involved in anomalous cancellation in the proper fraction A292288(n)/A292289(n).
%C A292393 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.
%C A292393 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
%H A292393 Michael De Vlieger, <a href="/A292393/b292393.txt">Table of n, a(n) for n = 2..120</a>
%H A292393 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/AnomalousCancellation.html">Anomalous Cancellation</a>
%F A292393 a(p) = 1.
%F A292393 a(p + 1) = p.
%e A292393 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.
%e A292393 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.
%e A292393 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.
%e A292393 Table relating a(n) with A292288(n) and A292289(n).
%e A292393 n = base and index.
%e A292393 N = A292288(n) = smallest numerator that pertains to D.
%e A292393 D = A292289(n) = smallest denominator that has a nontrivial anomalous cancellation in base n.
%e A292393 n/d = simplified ratio of numerator N and denominator D.
%e A292393 k = a(n) = base-n digit anomalously canceled in the numerator and denominator to arrive at N/D.
%e A292393 .
%e A292393 n N D N/D k
%e A292393 ------------------------------
%e A292393 2 3 6 1/2 1
%e A292393 3 4 12 1/3 1
%e A292393 4 7 14 1/2 3
%e A292393 5 6 30 1/5 1
%e A292393 6 11 33 1/3 5
%e A292393 7 8 56 1/7 1
%e A292393 8 15 60 1/4 7
%e A292393 9 13 39 1/3 4
%e A292393 10 16 64 1/4 6
%e A292393 11 12 132 1/11 1
%e A292393 12 23 138 1/6 11
%e A292393 13 14 182 1/13 1
%e A292393 14 27 189 1/7 13
%e A292393 15 22 110 1/5 7
%e A292393 16 21 84 1/4 5
%e A292393 17 18 306 1/17 1
%e A292393 18 35 315 1/9 17
%e A292393 19 20 380 1/19 1
%e A292393 20 39 390 1/10 19
%t A292393 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 *)
%Y A292393 Cf. A292288 (numerators), A292289 (denominators).
%K A292393 nonn,base
%O A292393 2,3
%A A292393 _Michael De Vlieger_, Sep 15 2017