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A213897 Fixed points of a sequence h(n) defined by the minimum number of 8's in the relation n*[n,8,8,...,8,n] = [x,...,x] between simple continued fractions.

%I A213897 #27 Sep 17 2015 04:40:47
%S A213897 3,7,23,31,71,107,131,139,163,199,211,227,283,347,367,379,419,431,439,
%T A213897 487,499,503,547,571,607,619,643,691,719,751,787,811,823,827,907,911,
%U A213897 983,991,1031,1051,1091,1151,1163,1231,1303,1319,1367,1399,1423,1439,1459,1499
%N A213897 Fixed points of a sequence h(n) defined by the minimum number of 8's in the relation n*[n,8,8,...,8,n] = [x,...,x] between simple continued fractions.
%C A213897 In a variant of A213891, multiply n by a number with simple continued fraction [n,8,8,..,8,n] and increase the number of 8's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are
%C A213897 2 * [2, 8, 2] = [4, 4, 4],
%C A213897 3 * [3, 8, 8, 8, 3] = [9, 2, 1, 2, 2, 2, 1, 2, 9],
%C A213897 4 * [4, 8, 4] = [16, 2, 16],
%C A213897 5 * [5, 8, 8, 5] = [25, 1, 1, 1, 1, 1, 1, 25],
%C A213897 6 * [6, 8, 8, 8, 6] = [36, 1, 2, 1, 4, 1, 2, 1, 36],
%C A213897 7 * [7, 8, 8, 8, 8, 8, 8, 8, 7] = [49, 1, 6, 4, 3, 2, 1, 2, 1, 2, 3, 4, 6, 1, 49].
%C A213897 The number of 8's needed defines the sequence h(n) = 1, 3, 1, 2, 3, 7, 1, 11, 5,.. (n>=2).
%C A213897 The current sequence contains the fixed points of h, i. e., those n where h(n)=n.
%C A213897 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)=8*f(n-1)+f(n-2), A041025, A015454, etc. This would mean that a prime is in the sequence A213897 if and only if it divides some term in each of the sequences satisfying f(n)=8*f(n-1)+f(n-2).
%C A213897 The sequence h() is recorded as A262218. - _M. F. Hasler_, Sep 15 2015
%t A213897 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[8, #] == # &] (* _Michael De Vlieger_, Sep 16 2015 *)
%o A213897 (PARI) {a(n) = local(t, m=1); if( n<2, 0, while( 1,
%o A213897    t = contfracpnqn( concat([n, vector(m,i,8), n]));
%o A213897    t = contfrac(n*t[1,1]/t[2,1]);
%o A213897    if(t[1]<n^2 || t[#t]<n^2, m++, break));
%o A213897 m)};
%o A213897 for(k=1,1500,if(k==a(k),print1(a(k),", ")));
%Y A213897 Cf. A000057, A213891 - A213896, A213898, A213899, A261311; A213358.
%Y A213897 Cf. A213648, A262212 - A262220, A213900, A262211.
%K A213897 nonn
%O A213897 1,1
%A A213897 _Art DuPre_, Jun 24 2012