A291094 -id:A291094 - cp's OEIS frontend

A367206 Denominators that appear in A291094 exactly twice.

160, 190, 196, 220, 231, 238, 242, 253, 260, 264, 275, 286, 291, 297, 303, 306, 309, 325, 390, 469, 536, 590, 591, 596, 597, 603, 609, 756, 790, 830, 897, 903, 906, 931, 960, 965, 996, 1014, 1040, 1060, 1064, 1100, 1120, 1160, 1162, 1169, 1190, 1191, 1197

Offset: 1

Jon E. Schoenfield, Nov 10 2023

nonn

See A291094 for a definition of anomalous cancellation.

			160 appears as the denominator in two unreduced fractions that can be reduced by anomalous cancellation: cancelling the 6s in the fractions 16/160 and 64/160 reduces them to 1/10 and 4/10, respectively.

Cf. A291094, A367207.

A367207 Denominators that appear in A291094 exactly three times.

Original entry on oeis.org

294, 330, 340, 341, 352, 363, 374, 385, 396, 404, 408, 640, 650, 670, 737, 796, 804, 894, 950, 970, 975, 985, 1015, 1030, 1196, 1204, 1208, 1210, 1265, 1300, 1328, 1330, 1463, 1494, 1604, 1660, 1670, 1728, 1837, 1920, 1952, 1958, 1975, 1985, 1988, 1996, 2004

Offset: 1

Jon E. Schoenfield, Nov 10 2023

nonn

See A291094 for a definition of anomalous cancellation.

			294 appears as the denominator in three unreduced fractions that can be reduced by anomalous cancellation: cancelling the 9s in the fractions 49/294, 98/294, and 196/294 reduces them to 4/24, 8/24, and 16/24, respectively.

Cf. A291094, A367206.

A291093 Form the list of fractions with nontrivial anomalous cancellation, sorted first by denominators, then by numerators; sequence lists the numerators.

Original entry on oeis.org

16, 26, 19, 49, 11, 12, 22, 13, 33, 34, 14, 44, 15, 55, 16, 64, 66, 17, 77, 18, 88, 19, 95, 96, 97, 39, 49, 98, 99, 101, 102, 103, 104, 21, 22, 121, 23, 33, 132, 34, 136, 24, 44, 143, 25, 55, 154, 26, 65, 66, 165, 106, 67, 27, 77, 176, 28, 88, 187, 29

Offset: 1

N. J. A. Sloane, Aug 21 2017

An unreduced fraction N/D is said to have the anomalous cancellation property if there is a single digit that can be cancelled from both N and D without changing the value of the fraction. The first and most famous example is 16/64 = 1/4 after cancelling the 6's.
Nontrivial means that fractions of the form x0/y0 are excluded (otherwise there would be a large number of trivial entries like 120/340).
The fractions are assumed to be in the range 0 to 1, and of course are not reduced.
The denominators d are considered in the order 11, 12, 13, ..., and then the numerators are considered in the order n = 10, 11, 12, ..., d-1.
A fraction is listed only once, even if the cancellation is possible in more than one way.

			The first two terms correspond to the fractions 16/64 = 1/4 (cancel the 6!) and 26/65 = 2/5 (again cancel the 6!).
The first 20 fractions are (before cancellation) 16/64, 26/65, 19/95, 49/98, 11/110, 12/120, 22/121, 13/130, 33/132, 34/136, 14/140, 44/143, 15/150, 55/154, 16/160, 64/160, 66/165, 17/170, 77/176, 18/180, which equal (after cancellation) 1/4, 2/5, 1/5, 1/2, 1/10, 1/10, 2/11, 1/10, 1/4, 1/4, 1/10, 4/13, 1/10, 5/14, 1/10, 2/5, 2/5, 1/10, 7/16, 1/10.

R. P. Boas, "Anomalous Cancellation." Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113-129, 1979.
A. Moessner, Scripta Math. 19; 20.
C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory. New York: Dover, 1988, pp. 86-87.

Michael De Vlieger, Table of n, a(n) for n = 1..8544 (numerators for denominators d <= 10^4; first 169 terms from N. J. A. Sloane)
Michael De Vlieger, Correlation of A291093 and A291094 and their ratio (all denominators d <= 10^4)
B. L. Schwartz, Proposal 434, Mathematics Magazine Vol. 34, No. 3 (1961), Problems and Questions, p. 173.
N. J. A. Sloane, Maple program.
N. J. A. Sloane, List of first 169 fractions (file gives line number, numerator, denominator).
Eric W. Weisstein, Anomalous Cancellation.

See A291094 for denominators.

Cf. A159975/A159976, A290462/A290463, A291966/A291965.

Flatten@ Table[Select[Range[11, 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@ Delete[x, #] &, Position[x, #]] &, Intersection @@ {n, d}]], {n, d}], -1]], # == Divide @@ {k, m} &]] @@ {k/m, #, First@ #, Last@ #} &@ Map[IntegerDigits, {k, m}] - Boole[Mod[{k, m}, 10] == {0, 0}]]], {m, 290}] (* Michael De Vlieger, Sep 13 2017 *)

A159975 Numerators (with multiplicity) of proper solutions up to 3-digit denominators of fractions with anomalous cancellation.

Original entry on oeis.org

13, 16, 19, 26, 124, 127, 138, 139, 145, 148, 154, 161, 163, 166, 176, 182, 187, 187, 187, 199, 218, 266, 273, 275, 286, 316, 327, 364, 412, 436

Offset: 1

Jonathan Vos Post, Apr 28 2009

The set of all proper solutions up to 3-digit denominators is given by 13/325, 16/64, 19/95, 26/65, 124/217, 127/762, 138/184, 139/973, 145/435, 148/185, 154/253, 161/644, 163/326, 166/664, 176/275, 182/819, 187/286, 187/385, 187/748, 199/995, 218/981, 266/665, 273/728, 275/374, 286/385, 316/632, 327/872, 364/637, 412/721, and 436/763.

			The first four values are the only four such cases for numerator and denominators of two digits: a(1) = 13 because 13/325 if you strike/cancel a digit "3" in numerator and denominator yields the correct 1/25. a(2) = 16 because 16/64 if you cancel a digit "6" in numerator and denominator yields the correct 1/4. a(3) = 19 because 19/95 if you cancel a digit "9" in numerator and denominator yields the correct 1/5.

Boas, R. P. "Anomalous Cancellation." Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113-129, 1979.
Moessner, A. Scripta Math. 19; 20.
Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 86-87, 1988.

Shalosh B. Ekhad, Automated Generation of Anomalous Cancellations, arXiv:1709.03379 [math.HO], 2017.
Eric W. Weisstein, Anomalous Cancellation.

Cf. A159976 (denominators)

For other fractions like this see A291093/A291094, A290462/A290463.

a(n)/A159976(n) is a proper fraction which undergoes Anomalous Cancellation.

A290462 Consider decimal fractions r = abc/def with b != 0, d != 0 such that r = ac/df, sorted first by def and then by abc; sequence gives the numerators abc.

Original entry on oeis.org

64, 95, 96, 97, 291, 98, 294, 490, 1221, 1342, 65, 665, 1463, 2261, 1584, 97, 194, 98, 196, 392, 490, 830, 195, 98, 196, 294, 490, 197, 591, 199, 597, 995, 1393, 1791, 2321, 1443, 2442, 3441, 1160, 1165, 2563, 3961, 1586

Offset: 1

N. J. A. Sloane, Aug 07 2017

These are "fractions with anomalous cancellation" of a particular type. Here the fractions are of the form abc/def, where the denominator has exactly three digits, such that if the tens digits (b and e) are canceled from the numerator and denominator the value is unchanged.
The numerator may have 2 or 3 digits.
For the denominators see A290463.
The full list of 171 terms is given in the a-file.

			The first four fractions on the list are 64/160 = 4/10 (after cancelling the 6's!), 95/190 = 5/10, 96/192 = 6/12, 97/194 = 7/14.

Doron Zeilberger, Email to N. J. A. Sloane, Aug 07 2017.

Doron Zeilberger, Table of n, a(n) for n = 1..171
Doron Zeilberger, The complete list of pairs [abc, def]

Cf. A290463.

For other fractions with anomalous cancellation see A291093/A291094, A159975/A159976.

A291965 Denominators of fractions with anomalous cancellation property, where more than one digit can be cancelled.

Original entry on oeis.org

64, 65, 95, 98, 121, 132, 136, 143, 154, 160, 165, 176, 184, 185, 187, 190, 192, 194, 195, 196, 198, 202, 206, 208, 217, 220, 231, 238, 242, 253, 260, 264, 265, 268, 275, 286, 291, 294, 297, 298, 303, 306, 309, 325, 326, 330, 332, 335, 340, 341, 345, 352, 363, 365, 374, 385, 390, 392, 394, 395, 396, 398, 404, 406, 408, 427, 435

Offset: 1

M. F. Hasler, Sep 06 2017

Here we refer to anomalous cancellation in a fraction if numerator and denominator have one or more digits in common, and the value of the fraction remains the same if all pairs of common digits are "cancelled", i.e., removed. (There are other variants of this definition, e.g., A291094, which differ in particular when there is more than one pair of common digits.)
For any solution one could add a trailing 0 to numerator and denominator and get another solution, but such solutions are excluded here.
See A291966 for the numerators. See the variant A291094 for other references.
The fractions are assumed to be between 0 and 1.

			The two-digit examples 16/64, 26/65, 19/95, 49/98 are well known. (The last one is particular in the sense that the (digit-wise) "simplified" fraction 4/8 is not in its lowest terms.)
The earliest three-digit terms correspond to 22/121 = 2/11, 33/132 = 3/12, 34/136 = 4/16, 44/143 = 4/13, 64/160 = 4/10, 55/154, 138/184 = 3/4 (first example of a two-digit simplification), ...
In constrast to A291093/A291094, the fraction 11/110 is not allowed here because we require that all pairs of common digits must be "cancelled", so the two 1's must be deleted, which leads to an invalid expression (with no digits left in the numerator).

M. F. Hasler, Table of n, N(n), a(n), for n = 1..146 (all terms a(n) < 1000), where N(n) = A291966(n) are the corresponding numerators.
B. L. Schwartz, Proposal 434, Mathematics Magazine Vol. 34, No. 3 (1961), Problems and Questions, p. 173.
Eric W. Weisstein, Anomalous Cancellation

Cf. A291966 (numerators), A291093/A291094 (alternate definition), A159975/A159976, A290462/A290463.

{is(n,dn=digits(n),Dn=Set(dn))=local(Cd,sc(x)=select(t->setsearch(Cd,t),x),rd(x)=local(S=0);fromdigits(select(d->!(setsearch(Cd,d)&&!bittest(S,d)&&S+=1<1 && #(Cd=setintersect(Set(dd=digits(d)),Dn)) && gcd(n,d)%10 ||next; rd(dd) || next; my(n1=rd(dn),d1=rd(dd),nd=digits(n1)); Cd=setintersect(Set(dd=digits(d1)),Set(nd)); if(#Cd, d*rd(nd)==n*rd(dd) && rd(dd), d*n1 == n*d1) && return(d))}

A291966 Numerators of fractions with the anomalous cancellation property, corresponding to denominators listed in A291965.

Original entry on oeis.org

16, 26, 19, 49, 22, 33, 34, 44, 55, 64, 66, 77, 138, 148, 88, 95, 96, 97, 39, 49, 99, 101, 103, 104, 124, 121, 33, 34, 44, 55, 65, 66, 106, 67, 77, 88, 97, 98, 99, 149, 101, 102, 206, 13, 163, 132, 83, 134, 136, 44, 138, 55, 66, 146, 77, 88, 195, 49, 197, 79, 99, 199, 101, 203, 102, 244, 145, 143, 55, 66, 116, 186, 67, 77, 88, 98, 99, 101

Offset: 1

M. F. Hasler, Sep 06 2017

See A291965 for more details, comments and references.

			The two-digit examples 16/64, 26/65, 19/95, 49/98 are well known.
The earliest three-digit terms of A291965 correspond to 34/136 = 4/16, 64/160 = 4/10, 138/184 = 3/4, ...

Cf. A291965, A291093/A291094, A159975/A159976, A290462/A290463.

/* Note: a(n) = A291966(A291965(n))! This function does not yield the n-th term, but the numerator corresponding to denominator N in A291965; if N is not in A291965, it yields zero. */ A291966(n, dn=digits(n), Dn=Set(dn))={local(Cd, sc(x)=select(t->setsearch(Cd, t), x), rd(x)=local(S=0); fromdigits(select(d->!(setsearch(Cd, d)&&!bittest(S, d)&&S+=1<1 && #(Cd=setintersect(Set(dd=digits(d)), Dn)) && gcd(n, d)%10 ||next; rd(dd) || next; my(n1=rd(dn), d1=rd(dd), nd=digits(n1)); Cd=setintersect(Set(dd=digits(d1)), Set(nd)); if(#Cd, d*rd(nd)==n*rd(dd) && rd(dd), d*n1 == n*d1) && return(d))}
/* To print this sequence: */ for(N=10,500,A291966(N)&&print1(A291966(N)","))

A292288 Numerators of smallest denominator of a proper fraction that has a nontrivial anomalous cancellation in base b.

Original entry on oeis.org

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

Offset: 2

Michael De Vlieger, Sep 15 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 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

			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

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, A292289 (denominators), A292393 (digit that is canceled).

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 *)

a(p) = (p + 1), for prime 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.

A366412 Number of nontrivial solutions to the P^*_k problem in base n.

Original entry on oeis.org

0, 0, 1, 0, 5, 0, 2, 2, 8, 0, 12, 0, 9, 13, 7, 0, 16, 0, 13, 19, 15, 0, 17, 6, 14, 6, 24, 0, 46, 0, 4, 18, 21, 15, 48, 0, 14, 18, 35, 0, 56, 0, 17, 36, 32, 0, 28, 10, 35, 33, 29, 0, 32, 31, 46, 33, 28, 0, 79, 0, 21, 31, 21, 25, 114, 0, 17, 30, 109, 0, 36, 0, 16, 40, 48, 28, 132

Offset: 2

Sayan Dutta, Oct 09 2023

For a given fixed base, the number [a_1 a_2 ... a_(2*k+1)] is said to satisfy the property P_k^* if [a_1 ... a_k]*[a_(k+1) ... a_(2*k+1)] = [a_1 ... a_(k+1)]*[a_(k+2) ... a_(2*k+1)], where [...] is to be interpreted as a block of digits.
The property P^*_k is a subcase of a mathematically inaccurate method where cancelling the common digits of the numerator and denominator correctly reduces it (usually known as anomalous cancellation). This sequence deals only with those anomalously cancellable fractions where there are equal numbers of digits in the numerator and denominator, and the last digit of the numerator is cancelled with the first digit of the denominator.
This is also equivalent to solving the Diophantine equation (a*B + b)*c = a*(b*B^k + c) with 0 < b < B and 0 < a,c < B^k.
All the solutions of a(p^n) where p^n is a prime power are three-digit solutions (proved in the paper by Saha et al.). For example, see Example section.
For a given base B, the number of solutions of P_k^* become constant beyond k=max{5, 2*log_2(B - 1) + 2} (proved in the paper by Saha et al.).
If [a_1 ... a_k b c_1 ... c_k] is a solution, then so is [a_1 ... a_k b b b c_1 ... c_k]. The latter is called an extension of the former, and is counted as a trivial solution. See Proposition 1 of Saha et al. link.
A solution is always of the form [a_1 ... a_k b...b c_k] (see Theorem 2 in the paper by Saha et al.).
It has been conjectured that for a given composite base B, if there are no new nontrivial solutions (except for extensions) in (2k + 1) digits, then there would be no new solutions in (2k + 3) digits (see Saha et al. link).

			For n = 10, the a(10) = 8 solutions correspond to 16/64 = 1/4, 26/65 = 2/5, 19/95 = 1/5, 49/98 = 4/8, 217/775 = 21/75, 249/996 = 24/96, 1249/9992 = 124/992 and 34027/77776 = 3402/7776.
For n = 9 = 3^2, the only two solutions are 14/43 and 28/86.

R. P. Boas, Anomalous Cancellation, The Two-Year College Mathematics Journal, Vol. 3, No. 2 (Autumn, 1972), pp. 21-24.
Shalosh B. Ekhad, Automated Generation of Anomalous Cancellations, arXiv:1709.03379 [math.HO], 2017.
Satvik Saha, Sohom Gupta, Sayan Dutta, and Sourin Chatterjee, Characterising Solutions of Anomalous Cancellation, arXiv:2302.00479 [math.HO], 2023.

Cf. A159975/A159976, A291093/A291094, A291965/A291966.

import math
LEN = 79
carr=[]
for base in range(2, LEN):
    k = int(2 * math.log(base - 1) / math.log(2) + 2) + 1
    k = max(k, 5)
    I = (base ** k - 1) // (base - 1) - 1
    count = 0
    for b in range(2, base):
        for c_k in range(1, b):
            c = b * I + c_k
            a = b * c // (b * base - (base - 1) * c_k)
            if a < base ** (k - 1):
                continue
            if (a * base + b) * c == a * (b * base**k + c):
                count += 1
    carr.append(count)
print(carr)

a(p) = 0 if and only if p is a prime (see Theorem 3 of Saha et al. link).

a(n) <= (n-2)*(n-3)/2 (see Proposition 4 of Saha et al. link).

cp's OEIS Frontend

A367206 Denominators that appear in A291094 exactly twice.

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A367207 Denominators that appear in A291094 exactly three times.

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A291093 Form the list of fractions with nontrivial anomalous cancellation, sorted first by denominators, then by numerators; sequence lists the numerators.

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Mathematica

A159975 Numerators (with multiplicity) of proper solutions up to 3-digit denominators of fractions with anomalous cancellation.

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A290462 Consider decimal fractions r = abc/def with b != 0, d != 0 such that r = ac/df, sorted first by def and then by abc; sequence gives the numerators abc.

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A291965 Denominators of fractions with anomalous cancellation property, where more than one digit can be cancelled.

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PARI

A291966 Numerators of fractions with the anomalous cancellation property, corresponding to denominators listed in A291965.

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PARI

A292288 Numerators of smallest denominator of a proper fraction that has a nontrivial anomalous cancellation in base b.

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

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