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A213895 Fixed points of a sequence h(n) defined by the minimum number of 6's in the relation n*[n,6,6,...,6,n] = [x,...,x] between simple continued fractions. 3
7, 11, 23, 47, 127, 139, 211, 223, 251, 331, 367, 379, 383, 463, 487, 499, 607, 619, 691, 727, 739, 743, 811, 823, 863, 887, 967, 971, 983, 1051, 1063, 1087, 1171, 1291, 1303, 1327, 1367, 1423, 1447, 1451, 1459 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

In a variant of A213891, multiply n by a number with simple continued fraction [n,6,6,..,6,n] and increase the number of 6's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are

2 * [2, 6, 2] = [4, 3, 4] ,

3 * [3, 6, 3] = [9, 2, 9],

4 * [4, 6, 6, 6, 4] = [16, 1, 1, 1, 5, 1, 1, 1, 16],

5 * [5, 6, 6, 6, 6, 5] = [25, 1, 4, 3, 3, 4, 1, 25],

6 * [6, 6, 6] = [36, 1, 36],

7 * [7, 6, 6, 6, 6, 6, 6, 6, 7] = [50, 7, 2, 1, 4, 4, 4, 1, 2, 7, 50].

The number of 6's needed defines the sequence h(n) = 1, 1, 3, 4, 1, 7, 7, 5, 9,... (n>=2).

The current sequence contains the fixed points of h, id est, those n where h(n)=n.

We conjecture that this sequence contains numbers is analogous to the sequence of prime numbers A000057, in the sense that, instead of referring to the Fibonacci sequences (sequences satisfying f(n)=f(n-1)+f(n-2) with arbitrary positive integer values for f(1) and f(2)) it refers to the generalized Fibonacci sequences satisfying f(n)=6*f(n-1)+f(n-2), A005668, A015451, A179237 etc. This would mean that a prime is in the sequence if and only if it divides some term in each of the sequences satisfying f(n)=6*f(n-1)+f(n-2).

The above sequence h() is recorded as A262216. - M. F. Hasler, Sep 15 2015

LINKS

Table of n, a(n) for n=1..41.

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]; Select[Range[2, 1000], f[6, #] == # &] (* Michael De Vlieger, Sep 16 2015 *)

PROG

(PARI)

{a(n) = local(t, m=1); if( n<2, 0, while( 1,

   t = contfracpnqn( concat([n, vector(m, i, 6), n]));

   t = contfrac(n*t[1, 1]/t[2, 1]);

   if(t[1]<n^2 || t[#t]<n^2, m++, break));

m)};

for(k=1, 1500, if(k==a(k), print1(a(k), ", ")));

CROSSREFS

Cf. A000057, A213891 - A213894, A213896 - A213899, A261311; A213358.

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

Sequence in context: A319135 A163848 A111671 * A140111 A118072 A141305

Adjacent sequences:  A213892 A213893 A213894 * A213896 A213897 A213898

KEYWORD

nonn

AUTHOR

Art DuPre, Jun 23 2012

STATUS

approved

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Last modified January 20 14:02 EST 2020. Contains 331094 sequences. (Running on oeis4.)