A213896 - OEIS

%I #22 Sep 16 2015 09:23:20

%S 2,3,19,31,67,79,103,127,139,151,167,179,191,263,283,359,383,443,463,

%T 479,491,503,571,631,691,787,827,883,919,1019,1087,1171,1291,1303,

%U 1307,1327,1399,1423,1451,1487

%N Fixed points of a sequence h(n) defined by the minimum number of 7's in the relation n*[n,7,7,...,7,n] = [x,...,x] between simple continued fractions.

%C In a variant of A213891, multiply n by a number with simple continued fraction [n,7,7,..,7,n] and increase the number of 7'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, 7, 7, 2] = [4, 3, 1, 1, 3, 4],

%C 3 * [3, 7, 7, 7, 3] = [9, 2, 2, 1, 1, 1, 2, 2, 9] ,

%C 4 * [4, 7, 7, 7, 7, 7, 4] = [16, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 16],

%C 5 * [5, 7, 7, 5] = [25, 1, 2, 2, 1, 25] ,

%C 6 * [6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6] = [36, 1, 5, 3, 1, 4, 10, 1, 2, 2, 4, 2, 2, 1, 10, 4, 1, 3, 5, 1, 36],

%C 7 * [7, 7, 7] = [49, 1, 49] .

%C The number of 7's needed defines the sequence h(n) = 2, 3, 5, 2, 11, 1, 5, 11, 2,...  (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)=7*f(n-1)+f(n-2), A054413, A015453, etc. This would mean that a prime is in the sequence A213896 if and only if it divides some term in each of the sequences satisfying f(n)=7*f(n-1)+f(n-2).

%C The above sequence h() is recorded as A262217. - _M. F. Hasler_, Sep 15 2015

%o (PARI)

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

%o    t = contfracpnqn( concat([n, vector(m,i,7), 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=1,1500,if(k==a(k),print1(a(k),", ")));

%Y Cf. A000057, A213891 - A213895, A213897 - A213899, A261311; A213358.

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

%K nonn

%O 1,1

%A _Art DuPre_, Jun 23 2012