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A213896 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. 4
2, 3, 19, 31, 67, 79, 103, 127, 139, 151, 167, 179, 191, 263, 283, 359, 383, 443, 463, 479, 491, 503, 571, 631, 691, 787, 827, 883, 919, 1019, 1087, 1171, 1291, 1303, 1307, 1327, 1399, 1423, 1451, 1487 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

2 * [2, 7, 7, 2] = [4, 3, 1, 1, 3, 4],

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

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

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

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],

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

The number of 7's needed defines the sequence h(n) = 2, 3, 5, 2, 11, 1, 5, 11, 2,...  (n>=2).

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

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).

The above sequence h() is recorded as A262217. - M. F. Hasler, Sep 15 2015

LINKS

Table of n, a(n) for n=1..40.

PROG

(PARI)

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

   t = contfracpnqn( concat([n, vector(m, i, 7), n]));

   t = contfrac(n*t[1, 1]/t[2, 1]);

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

m)};

for(k=1, 1500, if(k==a(k), print1(a(k), ", ")));

CROSSREFS

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

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

Sequence in context: A058912 A040145 A142955 * A088790 A283186 A215304

Adjacent sequences:  A213893 A213894 A213895 * A213897 A213898 A213899

KEYWORD

nonn

AUTHOR

Art DuPre, Jun 23 2012

STATUS

approved

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Last modified November 26 16:58 EST 2020. Contains 338641 sequences. (Running on oeis4.)