A262211 Minimum number of 12's such that n*[n; 12, ..., 12, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.

1, 1, 1, 2, 1, 5, 3, 5, 5, 9, 1, 6, 5, 5, 7, 8, 5, 19, 5, 5, 9, 23, 3, 14, 13, 17, 5, 2, 5, 31, 15, 9, 17, 5, 5, 36, 19, 13, 11, 19, 5, 43, 9, 5, 23, 45, 7, 5, 29, 17, 13, 12, 17, 29, 11, 19, 5, 59, 5, 30, 31, 5, 31, 20, 9, 65, 17, 23, 5, 13, 11, 3, 73, 29, 19, 29, 13, 79, 23, 53, 19, 81, 5, 8, 43, 5, 19, 14, 5, 41, 23, 31, 45, 59, 15, 48, 5, 29

Offset: 2

Author

M. F. Hasler, Sep 15 2015

Keywords

• nonn

Comments

Sequence A261311 lists fixed points of this sequence.

It is surprising that the variant A213900 with 11 instead of 12 has the same fixed points A000057 as the variant A213648 with 1 instead of 12, but other variants (A262212 - A262220 and this one) have different sets of fixed points (A213891 - A213899 and A261311).

Crossrefs

Cf. A213648, A262212 - A262220, A213900; A000057, A213891 - A213899.

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[12, #] & /@ 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}
A262211(n,d=12)=for(k=1,9e9,(c=contfrac(cf(vector(k+2,i,if(i>1&&i