bibkey: pongsriiam2021quasi authors: Prapanpong Pongsriiam year: 2021 title: “Quasi-Injectivity of Some Arithmetic Functions” doi: null url: https://cs.uwaterloo.ca/journals/JIS/VOL24/Pongsriiam/pong23.pdf claim: “Question 17 asks whether, for each m at least two, some function is quasi-injective of order m-1 but not of order m.” strata_touched:

D5/S3/Combinatorics/QuasiInjectiveOrderSeparation
D5/S3/Combinatorics/QuasiInjectiveCompositionRefutation license: citation-only triage: anchor

Pongsriiam quasi-injectivity of arithmetic functions

Verified locator

URL: https://cs.uwaterloo.ca/journals/JIS/VOL24/Pongsriiam/pong23.pdf

DOI: none assigned; Journal of Integer Sequences 24 (2021), Article 21.10.1. The journal does not mint a DOI for this article and no preprint version is listed on the author’s publication page, so the journal PDF is the locator.

Definitions

Definition 1 reads:

We call a function f : N → C a quasi-injective function if for all a, b ∈ N, the condition f(an) = f(bn) for all n ∈ N implies a = b. In addition, if f : N → N and ℓ ∈ N, then we say that f is quasi-injective of order ℓ if f, f^(2), f^(3), …, f^(ℓ) are quasi-injective, that is, for any a, b, k ∈ N with 1 ≤ k ≤ ℓ, if f^(k)(an) = f^(k)(bn) for all n ∈ N, then a = b.

Here f^(k) is the k-fold composite and N is the set of positive integers. The article notes immediately that quasi-injectivity of order ℓ implies quasi-injectivity of every order m ≤ ℓ, so the orders form a decreasing chain of conditions.

What the article settles

τ = σ_0 is quasi-injective of every order (Theorem 7), and the same holds for the Jordan totient functions J_s. For the divisor-power sums σ_s the article proves quasi-injectivity of order two and no further; Question 16 asks about the higher orders. So none of the functions studied there separates two consecutive orders, which is what Question 17 asks for.

The statement in question

Section 5 states:

Question 17. For each m ≥ 2, is there a function f : N → N such that f is quasi-injective of order m − 1 but not of order m?

Scope of the recorded answer

Yes, for every m ≥ 2. Give an odd number 2t + 1 with t ≥ 1 the level t mod (m − 1), and let

g_m(1)     = 1
g_m(2t)    = 2(m − 1)t + 1                               (t ≥ 1)
g_m(2t+1)  = 1        if t mod (m − 1) = (m − 2) mod (m − 1)
g_m(2t+1)  = 2t + 3   otherwise.

So g_m raises the level by one, sends the top level to 1, fixes 1, and sends an even number into level zero of the ladder. For 1 ≤ k ≤ m − 1 and t ≥ 1 the k-fold composite satisfies

g_m^(k)(2t) = 2((m − 1)t + k − 1) + 1 ,

which is strictly increasing in t, so n = 2 separates any two distinct multipliers and g_m is quasi-injective of order m − 1. Every input reaches 1 within m steps, so g_m^(m) is constantly 1; taking a = 1 and b = 2 shows quasi-injectivity of order m fails.

Entering the ladder only through the even numbers is the load-bearing choice. Reading the level off the exponent of two in n instead breaks at order one already: the multipliers a = 2^{m−1} and b = 2^m then give g^(k)(an) = g^(k)(bn) = 1 for every n and every k ≥ 1.

Bounded prior-resolution evidence

The journal article and its HTML record were opened. The author’s publication page lists this as the only item on quasi-injectivity and shows no follow-up. The one paper in the screened corpus that cites it, Pongsriiam, J. Integer Seq. 26 (2023), Article 23.9.1, cites it only in the reference list and does not treat orders of quasi-injectivity. Web search for the phrase “quasi-injective of order” returns only the unrelated module-theoretic and order-theoretic senses. Citation-index result pages were not reachable, so this is a bounded negative finding and no worldwide priority claim is made.

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