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A262219 Minimum number of 9's such that n*[n; 9, ..., 9, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.

%I A262219 #11 Sep 17 2015 04:38:59
%S A262219 2,1,5,4,5,5,5,1,14,11,5,6,5,9,11,16,5,17,29,5,11,21,5,24,20,5,5,14,
%T A262219 29,31,23,11,50,29,5,17,17,13,29,2,5,43,11,9,65,47,11,41,74,33,41,26,
%U A262219 5,59,5,17,14,57,29,30,95,5,47,34,11,67,101,21,29,7,5,35,17,49,17,11,41,79,59,17,2,3,5,84,131,29,11,43,29,41,65,31,47,89,23,7,41
%N A262219 Minimum number of 9's such that n*[n; 9, ..., 9, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.
%C A262219 Sequence A213898 lists fixed points of this sequence.
%t A262219 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[9, #] & /@ Range[2, 120] (* _Michael De Vlieger_, Sep 16 2015 *)
%o A262219 (PARI) cf(v)={t=v[#v];forstep(i=#v-1,1,-1,t=v[i]+1/t);t}
%o A262219 A262219(n,d=9)=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 A262219 Cf. A213648, A262212 - A262220, A213900, A262211.
%Y A262219 Cf. A000057, A213891 - A213899, A261311: fixed points of the above.
%K A262219 nonn
%O A262219 2,1
%A A262219 _M. F. Hasler_, Sep 15 2015