Increasing-decreasing patterns in the iteration of an arithmetic function

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Abstract: Let

O m e g a

be a set of positive integers and let

S : O m e g a i g h t a r r o w O m e g a

be an arithmetic function. Let

V = (v_{i})_{i = 1}^{n}

be a finite sequence of positive integers. An integer

m i n O m e g a

has extit{increasing-decreasing pattern}

V

with respect to

S

if, for all odd integers

i i n 1, l d o t s, n

, [ S^{v_1+ cdots + v_{i-1}}(m) < S^{v_1+ cdots + v_{i-1}+1}(m) < cdots < S^{v_1+ cdots + v_{i-1}+v_{i}}(m) ] and, for all even integers

i i n 2, l d o t s, n

, [ S^{v_1+ cdots + v_{i-1}}(m) > S^{v_1+ cdots +v_{i-1}+1}(m) > cdots > S^{v_1+ cdots +v_{i-1}+v_i}(m). ] The arithmetic function

S

is extit{wildly increasing-decreasing} if, for every finite sequence

V

of positive integers, there exists an integer

m i n O m e g a

such that

m

has increasing-decreasing pattern

V

with respect to

S

. This paper gives a proof that the Syracuse function is wildly increasing-decreasing.

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