bibkey: oeis2024a290322 authors: OEIS Foundation Inc. year: 2024 title: OEIS A290322 doi: null url: https://oeis.org/A290322 claim: A comment on OEIS A290322 conjectures that the sum of admissible units modulo n is nonzero whenever five divides n. strata_touched:

• D5/S3/Arith/CyclotomicFiveResidueSum license: citation-only triage: anchor

OEIS A290322

The sequence records, for each modulus from two onward, the residue modulo that modulus of the sum of the units below it whose fifth cyclotomic value is also a unit. A comment by Robert Israel dated January 23, 2024 conjectures that this residue is nonzero whenever five divides the modulus.

The admissible set

A residue counts when it is coprime to the modulus and the value of u^4 + u^3 + u^2 + u + 1 at it is coprime to the modulus as well. The companion sequence A290309 counts these residues and is marked easy and multiplicative; it supplies the count for prime moduli. The conjecture here concerns the first moment of the same set, weighted by the residue itself, which the companion does not address.

The caller checked the page on September 9, 2026 and found the statement still presented as a conjecture.

What the Lean module proves

D5/S3/Arith/CyclotomicFiveResidueSum proves the conjecture and, on the way, a sharper local statement than the conjecture asks for. The conjecture only asserts that the residue is nonzero; the module establishes the exact five-adic valuation of the sum, which is one less than the valuation of the modulus. Nonvanishing follows because that valuation is strictly smaller.

Two supporting results are of independent interest: the closed form of the sum over a prime power of five, and the fact that the count of admissible residues is never divisible by five when five does not divide the modulus.

ASSUMED-UNVERIFIED: the identification of the printed sequence with the module’s definition is a human reading of the entry, not a machine proof that the two denote the same function.

Search log

• Caller reading, 2026-09-09: the seat reported exact-phrase searches on the A-number, on the A-number with the words proof and with the author name, and on the entry’s own description together with the cyclotomic expression. It also reported checking the companion counting sequence A290309, and the polynomial-totient literature reached through Csizmazia and Toth (arXiv:2508.19103v2), whose Theorem 2.1 gives the multiplicativity and Euler product for polynomial-totient counting functions and whose Theorem 2.2 weights by a greatest common divisor rather than by the residue itself, so neither yields this weighted sum. It followed that paper’s citations back to Chidambaraswamy’s 1974 and 1979 papers on totients with respect to polynomials and found counting results only.
• The seat stated its receipt is 所查来源未找到 and not a claim that no proof exists anywhere; it has no access to restricted indexes.
• Caller verification, 2026-09-09: the orchestrator computed the sums directly and checked eight separate assertions before dispatching an implementation seat. The readings are recorded in the problem entry.

Verified locator

• URL: https://oeis.org/A290322
• Companion counting sequence: https://oeis.org/A290309