1,2

The fixed points of T^n are always rational of the form 2k/(2^n+-1) so the number of period n cycles is finite, and h(n) has this form.

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).

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; see section 5 h(m).

Orazio G. Cherubini, C++ program

a(2^n) = 2*A162634(n) for n>1 (empirical observation).

a(2n+1) = A020989(n) for n>1 (empirical observation).

For n=3: the cycles of period 3 in 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)=6.

For n=4: the cycles of period 4 in 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)=14.

Cf. A386237 (denominators), A385708 (binary expansion).

Cf. A162634, A020989.

Sequence in context: A200186 A370410 A192782 * A306742 A188576 A084214

Adjacent sequences: A385703 A385704 A385705 * A385707 A385708 A385709

nonn,more,frac

Orazio G. Cherubini, Jul 07 2025

approved