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

%I A291965 #14 Feb 16 2025 08:33:51
%S A291965 64,65,95,98,121,132,136,143,154,160,165,176,184,185,187,190,192,194,
%T A291965 195,196,198,202,206,208,217,220,231,238,242,253,260,264,265,268,275,
%U A291965 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
%N A291965 Denominators of fractions with anomalous cancellation property, where more than one digit can be cancelled.
%C A291965 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.)
%C A291965 For any solution one could add a trailing 0 to numerator and denominator and get another solution, but such solutions are excluded here.
%C A291965 See A291966 for the numerators. See the variant A291094 for other references.
%C A291965 The fractions are assumed to be between 0 and 1.
%H A291965 M. F. Hasler, <a href="/A291965/a291965_1.txt">Table of n, N(n), a(n)</a>, for n = 1..146 (all terms a(n) < 1000), where N(n) = A291966(n) are the corresponding numerators.
%H A291965 B. L. Schwartz, <a href="http://doi.org/10.2307/2688504">Proposal 434</a>, Mathematics Magazine Vol. 34, No. 3 (1961), Problems and Questions, p. 173.
%H A291965 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/AnomalousCancellation.html">Anomalous Cancellation</a>
%e A291965 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.)
%e A291965 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), ...
%e A291965 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).
%o A291965 (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))}
%Y A291965 Cf. A291966 (numerators), A291093/A291094 (alternate definition), A159975/A159976, A290462/A290463.
%K A291965 nonn,base,frac
%O A291965 1,1
%A A291965 _M. F. Hasler_, Sep 06 2017