cp's OEIS Frontend

A213898 Fixed points of a sequence h(n) defined by the minimum number of 9's in the relation n*[n,9,9,...,9,n] = [x,...,x] between simple continued fractions.

%I A213898 #30 Apr 27 2024 17:21:31
%S A213898 2,11,31,43,47,67,79,103,127,199,211,223,263,307,311,383,431,439,463,
%T A213898 467,499,523,563,571,587,691,719,751,811,839,863,883,911,967,991,1051,
%U A213898 1063,1087,1091,1123,1151,1231,1307,1327,1399,1447,1451,1459,1483,1499
%N A213898 Fixed points of a sequence h(n) defined by the minimum number of 9's in the relation n*[n,9,9,...,9,n] = [x,...,x] between simple continued fractions.
%C A213898 In a variant of A213891, multiply n by a number with simple continued fraction [n,9,9,..,9,n] and increase the number of 9's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are
%C A213898 2 * [2, 9, 9, 2] = [4, 4, 1, 1, 4, 4],
%C A213898 3 * [3, 9, 3] = [9, 3, 9],
%C A213898 4 * [4, 9, 9, 9, 9, 9, 4] = [16, 2, 3, 1, 1, 1, 1, 8, 1, 1, 1, 1, 3, 2, 16] ,
%C A213898 5 * [5, 9, 9, 9, 9, 5] = [25, 1, 1, 4, 1, 1, 1, 1, 1, 1, 4, 1, 1, 25],
%C A213898 6 * [6, 9, 9, 9, 9, 9, 6] = [36, 1, 1, 1, 13, 6, 13, 1, 1, 1, 36],
%C A213898 7 * [7, 9, 9, 9, 9, 9, 7] = [49, 1, 3, 3, 6, 1, 6, 3, 3, 1, 49].
%C A213898 The number of 9's needed defines the sequence h(n) = 2, 1,5, 4, 5, 5, 5, 1, 14,...  (n>=2).
%C A213898 The current sequence contains the fixed points of h, i. e., those n where h(n)=n.
%C A213898 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 sequence (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)=9*f(n-1)+f(n-2) like A099371, A015455 etc. This would mean that a prime is in the sequence A213898 if and only if it divides some term in each of the sequences satisfying f(n)=9*f(n-1)+f(n-2).
%t A213898 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[9, #] == # &] (* _Michael De Vlieger_, Sep 16 2015 *)
%o A213898 (PARI)
%o A213898 {a(n) = local(t, m=1); if( n<2, 0, while( 1,
%o A213898    t = contfracpnqn( concat([n, vector(m,i,9), n]));
%o A213898    t = contfrac(n*t[1,1]/t[2,1]);
%o A213898    if(t[1]<n^2 || t[#t]<n^2, m++, break));
%o A213898 m)};
%o A213898 for(k=1,1500,if(k==a(k),print1(a(k),", ")));
%Y A213898 Cf. A213358; A000057, A213891 - A213897, A213899, A261311.
%Y A213898 Cf. A213648, A262212 - A262220, A213900, A262211.
%K A213898 nonn
%O A213898 1,1
%A A213898 _Art DuPre_, Jun 24 2012