A213893 - OEIS

%I #38 Feb 10 2020 16:33:22

%S 3,7,43,67,103,127,163,223,283,367,463,487,523,547,607,643,727,787,

%T 823,883,907,1063,1123,1303,1327,1423,1447,1543,1567,1627,1663,1723,

%U 1747,1783,1867,1987,2083,2143,2203,2287,2347,2383,2467,2683,2707,2767,2803,2887

%N Fixed points of a sequence h(n) defined by the minimum number of 4's in the relation n*[n,4,4,...,4,n] = [x,...,x] between simple continued fractions.

%C In a variant of A213891, multiply n by a number with simple continued fraction [n,4,4,...,4,n] and increase the number of 4's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are

%C 2*[2,4,2] = [4,2,4],

%C 3*[3,4,4,4,3] = [9,1,2,2,2,1,9],

%C 4*[4,4,4] = [16,1,16],

%C 5*[5,4,4,4,4,5] = [26,5,1,1,5,26].

%C The number of 4's needed defines the sequence h(n) = 1, 3, 1, 4, 3, 7, 3, 3, 9, ... (n>=2).

%C The current sequence contains the fixed points of h, i.e., those n where h(n)=n.

%C 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) = 4*f(n-1) + f(n-2), A001076, A001077, A015448, 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) = 4*f(n-1) + f(n-2).

%C The above sequence h() is recorded as A262214. - _M. F. Hasler_, Sep 15 2015

%t 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[4, #] == # &] (* _Michael De Vlieger_, Sep 16 2015 *)

%o (PARI)

%o {a(n) = local(t, m=1); if( n<2, 0, while( 1,

%o    t = contfracpnqn( concat([n, vector(m,i,4), n]));

%o    t = contfrac(n*t[1,1]/t[2,1]);

%o    if(t[1]<n^2 || t[#t]<n^2, m++, break));

%o m)};

%o for(k=1,3000,if(k==a(k),print1(a(k),", ")));

%Y Cf. A000057, A213891, A213892, A213894 - A213899, A261311; A213358.

%Y Cf. A213648, A262212 - A262220, A213900, A262211.

%K nonn

%O 1,1

%A _Art DuPre_, Jun 23 2012