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A213894 Fixed points of a sequence h(n) defined by the minimum number of 5's in the relation n*[n,5,5,...,5,n] = [x,...,x] between simple continued fractions.

%I A213894 #38 Feb 10 2020 18:07:51
%S A213894 2,3,11,19,31,43,47,79,127,131,163,211,251,271,307,311,331,367,379,
%T A213894 443,503,563,599,607,659,743,751,823,839,859,887,907,911,947,967,1063,
%U A213894 1087,1091,1123,1163,1171,1187,1259,1279,1291,1303,1307,1319,1423,1447,1471,1487
%N A213894 Fixed points of a sequence h(n) defined by the minimum number of 5's in the relation n*[n,5,5,...,5,n] = [x,...,x] between simple continued fractions.
%C A213894 In a variant of A213891, multiply n by a number with simple continued fraction [n,5,5,...,5,n] and increase the number of 5's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are
%C A213894 2 * [2, 5, 5, 2] = [4, 2, 1, 1, 2, 4],
%C A213894 3 * [3, 5, 5, 5, 3] = [9, 1, 1, 2, 1, 2, 1, 1, 9],
%C A213894 4 * [4, 5, 5, 5, 5, 5, 4] = [16, 1, 3, 2, 1, 4, 1, 2, 3, 1, 16] ,
%C A213894 5 * [5, 5, 5] = [25, 1, 25].
%C A213894 The number of 5's needed defines the sequence h(n) = 2, 3, 5, 1, 11, 5, 5, 3, 5, 11, 11, ... (n >= 2).
%C A213894 The current sequence contains the fixed points of h, i.e., those n where h(n)=n.
%C A213894 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 generalized Fibonacci sequences satisfying f(n) = 5*f(n-1) + f(n-2), A052918, A015449, A164581, 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) = 5*f(n-1) + f(n-2).
%C A213894 The above sequence h() is recorded as A262215. - _M. F. Hasler_, Sep 15 2015
%t A213894 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[5, #] == # &] (* _Michael De Vlieger_, Sep 16 2015 *)
%o A213894 (PARI)
%o A213894 {a(n) = local(t, m=1); if( n<2, 0, while( 1,
%o A213894    t = contfracpnqn( concat([n, vector(m,i,5), n]));
%o A213894    t = contfrac(n*t[1,1]/t[2,1]);
%o A213894    if(t[1]<n^2 || t[#t]<n^2, m++, break));
%o A213894 m)};
%o A213894 for(k=1,1500,if(k==a(k),print1(a(k),", ")));
%Y A213894 Cf. A000057, A213891 - A213893, A213895 - A213899, A261311; A213358.
%Y A213894 Cf. A213648, A262212 - A262220, A213900, A262211.
%K A213894 nonn
%O A213894 1,1
%A A213894 _Art DuPre_, Jun 23 2012