A262214 Minimum number of 4's such that n*[n; 4, ..., 4, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.

1, 3, 1, 4, 3, 7, 3, 3, 9, 9, 3, 6, 7, 19, 7, 2, 3, 5, 9, 7, 9, 7, 3, 24, 13, 11, 7, 13, 19, 9, 15, 19, 5, 39, 3, 18, 5, 27, 19, 19, 7, 43, 9, 19, 7, 15, 7, 55, 49, 11, 13, 8, 11, 9, 7, 11, 13, 57, 19, 4, 9, 7, 31, 34, 19, 67, 5, 7, 39, 69, 3, 36, 37, 99, 5, 39, 27, 25, 39, 35, 19, 27, 7, 14, 43, 27, 19, 10, 19, 55, 7, 19, 15, 29, 15, 48, 55, 19

Offset

2,2

Comments

Sequence A213893 lists fixed points of this sequence.

Links

Table of n, a(n) for n=2..99.

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

Prog

(PARI) cf(v)={t=v[#v]; forstep(i=#v-1, 1, -1, t=v[i]+1/t); t}
A262214(n, d=4)=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))

Crossrefs

Cf. A213648, A262212 - A262220, A213900, A262211;

Cf. A000057, A213891 - A213899, A261311.

Sequence in context: A326041 A209613 A264596 * A340760 A035626 A082587

Adjacent sequences: A262211 A262212 A262213 * A262215 A262216 A262217

Keyword

nonn

Author

M. F. Hasler, Sep 15 2015

Status

approved