cp's OEIS Frontend

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.

%I A213892 #41 Feb 10 2020 18:07:30
%S A213892 2,7,19,31,47,67,71,83,151,163,167,223,227,271,307,331,359,379,431,
%T A213892 463,479,487,499,631,643,683,691,743,787,811,839,863,947,967,1019,
%U A213892 1051,1087,1103,1123,1163,1259,1279,1307,1319,1399,1423,1451,1471
%N 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.
%C A213892 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
%C A213892 2 * [2, 3, 3, 2] = [4, 1, 1, 1, 1, 4],
%C A213892 3 * [3, 3, 3] = [9, 1, 9],
%C A213892 4 * [4, 3, 3, 3, 3, 3, 4] = [17, 4, 1, 2, 1, 4, 17],
%C A213892 5 * [5, 3, 3, 5] = [26, 1, 1, 26],
%C A213892 6 * [6, 3, 3, 3, 3, 3, 6] = [37, 1, 4, 2, 4, 1, 37],
%C A213892 7 * [7, 3, 3, 3, 3, 3, 3, 3, 7] = [51, 8, 2, 1, 2, 8, 51].
%C A213892 The number of 3's needed defines the sequence h(n) = 2, 1, 5, 2, 5, 7, 5, 9, 2, ... (n>=2).
%C A213892 The current sequence contains the fixed points of h, i.e., those n where h(n)=n.
%C A213892 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).
%C A213892 The above sequence h() is recorded as A262213. - _M. F. Hasler_, Sep 15 2015
%t A213892 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 *)
%o A213892 (PARI)
%o A213892 {a(n) = local(t, m=1); if( n<2, 0, while( 1,
%o A213892    t = contfracpnqn( concat([n, vector(m,i,3), n]));
%o A213892    t = contfrac(n*t[1,1]/t[2,1]);
%o A213892    if(t[1]<n^2 || t[#t]<n^2, m++, break));
%o A213892 m)};
%o A213892 for(k=1,1500,if(k==a(k),print1(a(k),", ")));
%Y A213892 Cf. A000057, A213891, A213893 - A213899, A261311; A213358.
%Y A213892 Cf. A213648, A262212 - A262220, A213900, A262211.
%K A213892 nonn
%O A213892 1,1
%A A213892 _Art DuPre_, Jun 23 2012