A213897 -id:A213897 - cp's OEIS frontend

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.

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

Offset: 1

Art DuPre, Jun 23 2012

nonn

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

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

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

{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]

A213898 Fixed points of a sequence h(n) defined by the minimum number of 9's in the relation n*[n,9,9,...,9,n] = [x,...,x] between simple continued fractions.

Original entry on oeis.org

2, 11, 31, 43, 47, 67, 79, 103, 127, 199, 211, 223, 263, 307, 311, 383, 431, 439, 463, 467, 499, 523, 563, 571, 587, 691, 719, 751, 811, 839, 863, 883, 911, 967, 991, 1051, 1063, 1087, 1091, 1123, 1151, 1231, 1307, 1327, 1399, 1447, 1451, 1459, 1483, 1499

Offset: 1

Art DuPre, Jun 24 2012

nonn

In a variant of A213891, multiply n by a number with simple continued fraction [n,9,9,..,9,n] and increase the number of 9's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are
2 * [2, 9, 9, 2] = [4, 4, 1, 1, 4, 4],
3 * [3, 9, 3] = [9, 3, 9],
4 * [4, 9, 9, 9, 9, 9, 4] = [16, 2, 3, 1, 1, 1, 1, 8, 1, 1, 1, 1, 3, 2, 16] ,
5 * [5, 9, 9, 9, 9, 5] = [25, 1, 1, 4, 1, 1, 1, 1, 1, 1, 4, 1, 1, 25],
6 * [6, 9, 9, 9, 9, 9, 6] = [36, 1, 1, 1, 13, 6, 13, 1, 1, 1, 36],
7 * [7, 9, 9, 9, 9, 9, 7] = [49, 1, 3, 3, 6, 1, 6, 3, 3, 1, 49].
The number of 9's needed defines the sequence h(n) = 2, 1,5, 4, 5, 5, 5, 1, 14,...  (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 sequence (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)=9*f(n-1)+f(n-2) like A099371, A015455 etc. This would mean that a prime is in the sequence A213898 if and only if it divides some term in each of the sequences satisfying f(n)=9*f(n-1)+f(n-2).

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

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

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[9, #] == # &] (* Michael De Vlieger, Sep 16 2015 *)

{a(n) = local(t, m=1); if( n<2, 0, while( 1,
   t = contfracpnqn( concat([n, vector(m,i,9), n]));
   t = contfrac(n*t[1,1]/t[2,1]);
   if(t[1]

A262218 Minimum number of 8's such that n*[n; 8, ..., 8, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.

Original entry on oeis.org

1, 3, 1, 2, 3, 7, 1, 11, 5, 3, 3, 2, 7, 11, 3, 16, 11, 17, 5, 7, 3, 23, 3, 14, 5, 35, 7, 14, 11, 31, 7, 3, 33, 23, 11, 18, 17, 11, 5, 20, 7, 41, 3, 11, 23, 45, 3, 55, 29, 67, 5, 25, 35, 11, 7, 35, 29, 57, 11, 30, 31, 23, 15, 2, 3, 5, 33, 23, 23, 71, 11, 36, 37, 59, 17, 7, 11, 15, 11, 107, 41, 81, 7, 50, 41, 59, 3, 43, 11, 23, 23, 31, 45, 17, 7, 48, 55, 11

Offset: 2

M. F. Hasler, Sep 15 2015

nonn

Sequence A213897 lists fixed points of this sequence.

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

Cf. A000057, A213891 - A213899, A261311: fixed points of the above.

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]; f[8, #] & /@ Range[2, 120] (* Michael De Vlieger, Sep 16 2015 *)

cf(v)={t=v[#v];forstep(i=#v-1,1,-1,t=v[i]+1/t);t}
A262218(n,d=8)=for(k=1,9e9,(c=contfrac(cf(vector(k+2,i,if(i>1&&i

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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.

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A213898 Fixed points of a sequence h(n) defined by the minimum number of 9's in the relation n*[n,9,9,...,9,n] = [x,...,x] between simple continued fractions.

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A262218 Minimum number of 8's such that n*[n; 8, ..., 8, n] = [x; ..., x] for some x, where [...] denotes simple continued fractions.

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