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A262220 Minimum number of 10's such that n*[n; 10, ..., 10, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.

%I A262220 #8 Sep 17 2015 04:39:55
%S A262220 1,3,3,1,3,7,7,11,1,9,3,12,7,3,15,3,11,17,3,7,9,21,7,9,25,35,7,14,3,
%T A262220 31,31,19,3,7,11,8,17,51,7,20,7,43,19,11,21,47,15,55,9,3,51,8,35,9,7,
%U A262220 35,29,57,3,30,31,23,63,25,19,21,3,43,7,71,23,36,17,19,35,39,51,77,15,107,41,81,7,3,43,59,39,44,11,103,43,31,47,17,31,48,55,59
%N A262220 Minimum number of 10's such that n*[n; 10, ..., 10, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.
%C A262220 Sequence A213899 lists fixed points of this sequence.
%t A262220 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[10, #] & /@ Range[2, 120] (* _Michael De Vlieger_, Sep 16 2015 *)
%o A262220 (PARI) cf(v)={t=v[#v];forstep(i=#v-1,1,-1,t=v[i]+1/t);t}
%o A262220 A262220(n,d=10)=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 A262220 Cf. A213648, A213891 - A213899; A262212 - A262219, A213900.
%K A262220 nonn
%O A262220 2,2
%A A262220 _M. F. Hasler_, Sep 15 2015