cp's OEIS Frontend

A213899 Fixed points of a sequence h(n) defined by the minimum number of 10's in the relation n*[n,10,10,...,10,n] = [x,...,x] between simple continued fractions.

%I A213899 #33 Feb 10 2020 18:26:16
%S A213899 3,7,31,43,47,71,107,151,167,179,211,223,239,251,271,283,419,431,463,
%T A213899 467,487,491,523,547,563,571,631,839,859,883,907,967,971,1087,1103,
%U A213899 1171,1187,1279,1283,1291,1367,1399,1423,1459,1471,1483,1487,1499
%N A213899 Fixed points of a sequence h(n) defined by the minimum number of 10's in the relation n*[n,10,10,...,10,n] = [x,...,x] between simple continued fractions.
%C A213899 In a variant of A213891, multiply n by a number with simple continued fraction [n,10,10,...,10,n] and increase the number of 10's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are
%C A213899 2 * [2, 10, 2] = [4, 5, 4],
%C A213899 3 * [3, 10, 10, 10, 3] = [9, 3, 2, 1, 2, 1, 2, 3, 9],
%C A213899 4 * [4, 10, 10, 10, 4] = [16, 2, 1, 1, 9, 1, 1, 2, 16],
%C A213899 5 * [5, 10, 5] = [25, 2, 25],
%C A213899 6 * [6, 10, 10, 10, 6] = [36, 1, 1, 2, 6, 2, 1, 1, 36],
%C A213899 7 * [7, 10, 10, 10, 10, 10, 10, 10, 7] = [49, 1, 2, 3, 1, 6, 2, 1, 2, 2, 2, 1, 2, 6, 1, 3, 2, 1, 49].
%C A213899 The number of 10's needed defines the sequence h(n) = 1, 3, 3, 1, 3, 7, 7, 11, 1, ... (n>=2).
%C A213899 The current sequence contains the fixed points of h, i.e., those n where h(n)=n.
%C A213899 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) = 10*f(n-1) + f(n-2), A041041, A015456, etc. This would mean that a prime is in the sequence A213899 if and only if it divides some term in each of the sequences satisfying f(n) = 10*f(n-1) + f(n-2).
%C A213899 The sequence h() is given in A262220. - _M. F. Hasler_, Sep 15 2015
%t A213899 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[10, #] == # &] (* _Michael De Vlieger_, Sep 16 2015 *)
%o A213899 (PARI)
%o A213899 {a(n) = local(t, m=1); if( n<2, 0, while( 1,
%o A213899    t = contfracpnqn( concat([n, vector(m,i,10), n]));
%o A213899    t = contfrac(n*t[1,1]/t[2,1]);
%o A213899    if(t[1]<n^2 || t[#t]<n^2, m++, break));
%o A213899 m)};
%o A213899 for(k=1,1500,if(k==a(k),print1(a(k),", ")));
%Y A213899 Cf. A000057, A213891 - A213898, A261311.
%Y A213899 Cf. A213648, A262212 - A262220, A213900, A262211.
%K A213899 nonn
%O A213899 1,1
%A A213899 _Art DuPre_, Jun 24 2012