A291966 -id:A291966 - cp's OEIS frontend

A291094 Denominators of fractions with nontrivial anomalous cancellation, listed with multiplicity if multiple numerators are possible.

64, 65, 95, 98, 110, 120, 121, 130, 132, 136, 140, 143, 150, 154, 160, 160, 165, 170, 176, 180, 187, 190, 190, 192, 194, 195, 196, 196, 198, 202, 204, 206, 208, 210, 220, 220, 230, 231, 231, 238, 238, 240, 242, 242, 250

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 canceled from both N and D without changing the value of the fraction. The first and most famous example is 16/64 = 1/4 after canceling 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.
From Jon E. Schoenfield, Sep 12 2017: (Start)
For k = 1..12, the smallest denominator D that appears exactly k times and its corresponding numerators are as follows:
.
   k     D  numerators
  ==  ====  ================================================
   1    64  16
   2   160  16  64
   3   294  49  98 196
   4   392  49  98 196 294
   5   490  49  98 196 294 392
   6   660  66 165 264 363 462 561
   7   770  77 176 275 374 473 572 671
   8   880  88 187 286 385 484 583 682 781
   9   990  99 198 297 396 495 594 693 792 891
  10  1980  99 198 297 396 495 594 693 792 891 990
  11  2970  99 198 297 396 495 594 693 792 891 990 1980
  12  3960  99 198 297 396 495 594 693 792 891 990 1980 2970
Smallest denominator that appears exactly k times in the sequence for k = 1..41: 64, 160, 294, 392, 490, 660, 770, 880, 990, 1980, 2970, 3960, 4950, 5830, 6710, 7920, 8910, 9900, 11940, 12935, 13065, 14925, 15920, 16080, 16915, 18905, 19095, 23952, 24950, 25948, 26052, 24309, 28942, 29940, 29058, 31396, 32934, 34068, 33932, 35928, 36926 (note that this sequence is nonmonotonic; e.g., its 29th and 32nd terms are 24950 and 24309, respectively).
(End)

			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 (denominators d <= 10^4; first 169 terms from N. J. A. Sloane)
Michael De Vlieger, Correlation of A291093 and A291094 and their ratio (for 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 A291093 for numerators.
Cf. A159975/A159976, A290462/A290463.
Cf. A291965/A291966 for a variant.
Cf. A367206, A367207.

Flatten@ Table[ConstantArray[m, Count[Range[11, m - 1], ?(Function[k, Function[{r, 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, 250}] (* _Michael De Vlieger, Sep 13 2017 *)

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 *)

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))}

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).

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A291094 Denominators of fractions with nontrivial anomalous cancellation, listed with multiplicity if multiple numerators are possible.

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

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

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