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

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..67.

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

Example

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).

Prog

(PARI) {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<<d), x))); for(d=10, n-1, gcd(d, n)>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))}

Crossrefs

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

Sequence in context: A248770 A272297 A291094 * A255570 A255571 A151984

Adjacent sequences: A291962 A291963 A291964 * A291966 A291967 A291968

Keyword

nonn,base,frac

Author

M. F. Hasler, Sep 06 2017

Status

approved