A386237 Denominators of h(n) which is the minimum among the maxima of period n cycles of T(x) = 1 - 2 * |x-1/2|.

1, 5, 7, 17, 31, 63, 127, 257, 511, 1023, 2047, 1365, 8191, 16383, 32767, 65537, 131071, 262143, 524287, 349525, 2097151, 4194303, 8388607, 372827, 33554431

Offset

1,2

Comments

The fixed points of T^n are always rational of the form 2k/(2^n+-1) so the minimum among the maxima has again this property.

A truncated map can be formed T_h(x)=min(h,T(x)) and h(n) is the smallest h for which this map still has a period n cycle (falling between T_0 having only the fixed point 0, and T_1 which is all of T).

It appears that a(n)=A000225(n) for n not a power of 2 when a(n) does not simplify with the numerator.

It appears that a(2^n)=A000051(2^n)=A000215(n) when a(2^n) does not simplify with the denominator.

Links

Table of n, a(n) for n=1..25.

Keith Burns and Boris Hasselblatt, The Sharkovsky Theorem: A Natural Direct Proof, The American Mathematical Monthly, Vol. 118, No. 3 (2011), pp. 229-244; alternative link.

Orazio G. Cherubini, a386237

Example

For n=3: the three cycles of T are {2/7,4/7,6/7} and {2/9,4/9,8/9} with maxima 6/7 and 8/9. The minimum between those last numbers is 6/7 so a(3)=7.
For n=4: the four cycles of T are {2/15,4/15,8/15,14/15}, {2/17,4/17,8/17,16/17} and {6/17,12/17,10/17,14/17} with maxima 14/15,16/17,14/17. The minimum between those last numbers is 14/17 so a(4)=17.

Crossrefs

Cf. A385706 (numerators).

Cf. A385708 (periodic part of binary expansion of A385706(n)/a(n)).

Cf. A000215,A000051,A000225 (for empirical observations).

Sequence in context: A019340 A290471 A261792 * A166977 A272717 A018538

Adjacent sequences: A386234 A386235 A386236 * A386238 A386239 A386240

Keyword

nonn,frac,more

Author

Orazio G. Cherubini, Jul 16 2025

Status

approved