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A262217 Minimum number of 7's such that n*[n; 7, ..., 7, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.

%I A262217 #9 Sep 16 2015 15:26:02
%S A262217 2,3,5,2,11,1,5,11,2,9,11,5,5,11,11,3,11,19,5,3,29,7,11,2,5,35,5,6,11,
%T A262217 31,23,19,11,5,11,8,59,11,5,20,11,13,29,11,23,45,11,13,2,3,5,52,35,29,
%U A262217 5,19,20,57,11,30,95,11,47,5,59,67,11,7,5,23,11,36,8,11,59,9,11,79,11,107,20,27,11,11,41,27,29,43,11,5,23,31,137,59,23,47,41,59
%N A262217 Minimum number of 7's such that n*[n; 7, ..., 7, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.
%C A262217 Sequence A213896 lists fixed points of this sequence.
%t A262217 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[7, #] & /@ Range[2, 120] (* _Michael De Vlieger_, Sep 16 2015 *)
%o A262217 (PARI) cf(v)={t=v[#v];forstep(i=#v-1,1,-1,t=v[i]+1/t);t}
%o A262217 A262217(n,d=7)=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 A262217 Cf. A213648, A262212 - A262220, A213900, A262211.
%Y A262217 Cf. A000057, A213891 - A213899, A261311: fixed points of the above.
%K A262217 nonn
%O A262217 2,1
%A A262217 _M. F. Hasler_, Sep 15 2015