Divisibility Set of the Generalized Euler Totient

Define $\varphi_k(n) = \sum_{1 \le a \le n, (a,n)=1} a^k$ and $\mathcal{D}_s = \{k \ge s : \varphi_s(n) \mid \varphi_k(n) \text{ for every } n\}$. Is $\mathcal{D}_1 = \{1, 3, 15\}$, as conjectured by Büyükaşik and collaborators?

Result

Proved

Status

Resolved

AI contribution

AI co-developed

Method

Argument

Field

Multiplicative number theory

Posed by

Engin Büyükaşik et al.

Year posed

2024

Years open

2y

Solved

2026-06-01

Model

GPT-5.5 Pro

Vendor

OpenAI

Collaborators

John M. Campbell

Verification

Unreviewed

Publication

Preprint

Significance

5 / 100*

Disclosed cost

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Wikipedia

No dedicated article