cp's OEIS Frontend

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.

%I A213895 #33 Feb 10 2020 18:26:10
%S A213895 7,11,23,47,127,139,211,223,251,331,367,379,383,463,487,499,607,619,
%T A213895 691,727,739,743,811,823,863,887,967,971,983,1051,1063,1087,1171,1291,
%U A213895 1303,1327,1367,1423,1447,1451,1459
%N 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.
%C A213895 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
%C A213895 2 * [2, 6, 2] = [4, 3, 4],
%C A213895 3 * [3, 6, 3] = [9, 2, 9],
%C A213895 4 * [4, 6, 6, 6, 4] = [16, 1, 1, 1, 5, 1, 1, 1, 16],
%C A213895 5 * [5, 6, 6, 6, 6, 5] = [25, 1, 4, 3, 3, 4, 1, 25],
%C A213895 6 * [6, 6, 6] = [36, 1, 36],
%C A213895 7 * [7, 6, 6, 6, 6, 6, 6, 6, 7] = [50, 7, 2, 1, 4, 4, 4, 1, 2, 7, 50].
%C A213895 The number of 6's needed defines the sequence h(n) = 1, 1, 3, 4, 1, 7, 7, 5, 9, ... (n>=2).
%C A213895 The current sequence contains the fixed points of h, i.e., those n where h(n)=n.
%C A213895 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).
%C A213895 The above sequence h() is recorded as A262216. - _M. F. Hasler_, Sep 15 2015
%t A213895 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 *)
%o A213895 (PARI)
%o A213895 {a(n) = local(t, m=1); if( n<2, 0, while( 1,
%o A213895    t = contfracpnqn( concat([n, vector(m,i,6), n]));
%o A213895    t = contfrac(n*t[1,1]/t[2,1]);
%o A213895    if(t[1]<n^2 || t[#t]<n^2, m++, break));
%o A213895 m)};
%o A213895 for(k=1,1500,if(k==a(k),print1(a(k),", ")));
%Y A213895 Cf. A000057, A213891 - A213894, A213896 - A213899, A261311; A213358.
%Y A213895 Cf. A213648, A262212 - A262220, A213900, A262211.
%K A213895 nonn
%O A213895 1,1
%A A213895 _Art DuPre_, Jun 23 2012