A213892 Fixed points of a sequence h(n) defined by the minimum number of 3's in the relation n*[n,3,3,...,3,n] = [x,...,x] between simple continued fractions.

2, 7, 19, 31, 47, 67, 71, 83, 151, 163, 167, 223, 227, 271, 307, 331, 359, 379, 431, 463, 479, 487, 499, 631, 643, 683, 691, 743, 787, 811, 839, 863, 947, 967, 1019, 1051, 1087, 1103, 1123, 1163, 1259, 1279, 1307, 1319, 1399, 1423, 1451, 1471

Offset

1,1

Comments

In a variant of A213891, multiply n by a number with simple continued fraction [n,3,3,...,3,n] and increase the number of 3's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are

2 * [2, 3, 3, 2] = [4, 1, 1, 1, 1, 4],

3 * [3, 3, 3] = [9, 1, 9],

4 * [4, 3, 3, 3, 3, 3, 4] = [17, 4, 1, 2, 1, 4, 17],

5 * [5, 3, 3, 5] = [26, 1, 1, 26],

6 * [6, 3, 3, 3, 3, 3, 6] = [37, 1, 4, 2, 4, 1, 37],

7 * [7, 3, 3, 3, 3, 3, 3, 3, 7] = [51, 8, 2, 1, 2, 8, 51].

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

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

We conjecture that this sequence contains prime numbers 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 sequences satisfying f(n) = 3*f(n-1) + f(n-2), A006190, A003688, A052924, 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) = 3*f(n-1) + f(n-2).

The above sequence h() is recorded as A262213. - M. F. Hasler, Sep 15 2015

Links

Table of n, a(n) for n=1..48.

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[3, #] == # &] (* 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, 3), 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, A213893 - A213899, A261311; A213358.

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

Sequence in context: A179002 A232667 A034794 * A152608 A334520 A308269

Adjacent sequences: A213889 A213890 A213891 * A213893 A213894 A213895

Keyword

nonn

Author

Art DuPre, Jun 23 2012

Status

approved