A262212 Minimum number of 2's such that n*[n; 2, ..., 2, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.

1, 3, 3, 2, 3, 5, 7, 11, 5, 11, 3, 6, 5, 11, 15, 7, 11, 19, 11, 11, 11, 21, 7, 14, 13, 35, 11, 4, 11, 29, 31, 11, 7, 5, 11, 18, 19, 27, 23, 9, 11, 43, 11, 11, 21, 45, 15, 41, 29, 7, 27, 26, 35, 11, 23, 19, 9, 19, 11, 30, 29, 11, 63, 20, 11, 67, 7, 43, 5, 69, 23, 35, 37, 59, 19, 11, 27, 25, 47, 107, 9, 83, 11, 23, 43, 19, 23, 43, 11, 41, 43, 59, 45, 59, 31

Offset: 2

Author

M. F. Hasler, Sep 15 2015

Keywords

• nonn

Comments

Sequence A213891 lists fixed points of this sequence.

Crossrefs

Cf. A213648, A262213 - A262220, A213900, A262211; A000057, A213891 - A213899, A261311.

Programs

Mathematica

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[2, #] & /@ Range[2, 120] (* Michael De Vlieger, Sep 16 2015 *)

PARI

cf(v)={t=v[#v];forstep(i=#v-1,1,-1,t=v[i]+1/t);t}
A262212(n,d=2)=for(k=1,9e9,(c=contfrac(cf(vector(k+2,i,if(i>1&&i