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A262218 Minimum number of 8's such that n*[n; 8, ..., 8, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.

%I A262218 #8 Sep 17 2015 04:39:09
%S A262218 1,3,1,2,3,7,1,11,5,3,3,2,7,11,3,16,11,17,5,7,3,23,3,14,5,35,7,14,11,
%T A262218 31,7,3,33,23,11,18,17,11,5,20,7,41,3,11,23,45,3,55,29,67,5,25,35,11,
%U A262218 7,35,29,57,11,30,31,23,15,2,3,5,33,23,23,71,11,36,37,59,17,7,11,15,11,107,41,81,7,50,41,59,3,43,11,23,23,31,45,17,7,48,55,11
%N A262218 Minimum number of 8's such that n*[n; 8, ..., 8, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.
%C A262218 Sequence A213897 lists fixed points of this sequence.
%t A262218 f[m_, n_] := Block[{c, k = 1}, c[x_, y_] := ContinuedFraction[x FromContinuedFraction[Join[{x}, Table[m, {y}], {x}]]]; While[First@ c[n, k] != Last@ c[n, k], k++]; k]; f[8, #] & /@ Range[2, 120] (* _Michael De Vlieger_, Sep 16 2015 *)
%o A262218 (PARI) cf(v)={t=v[#v];forstep(i=#v-1,1,-1,t=v[i]+1/t);t}
%o A262218 A262218(n,d=8)=for(k=1,9e9,(c=contfrac(cf(vector(k+2,i,if(i>1&&i<k+2,d,n)))*n))[1]==c[#c]&&return(k))
%Y A262218 Cf. A213648, A262212 - A262220, A213900, A262211.
%Y A262218 Cf. A000057, A213891 - A213899, A261311: fixed points of the above.
%K A262218 nonn
%O A262218 2,2
%A A262218 _M. F. Hasler_, Sep 15 2015