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slug: oeis-a280258-krizek-phitorial-divisor-sum-parity bibkey: krizek2017a280258 doi: null url: https://oeis.org/A280258 triage: theorem motivation_gids:

  • D5/S3/Arith/KrizekPhitorialDivisorSumParity.result

Krizek’s phitorial divisor-sum parity characterization

Problem

OEIS A280258, NAME (%N, verbatim):

a(n) = Sum_{d|n} pxi(d), where pxi(m) is the product of totatives of m (A001783).

Jaroslav Krizek’s COMMENT (%C, verbatim):

Conjecture: a(n) is odd for numbers in A183300; a(n) is even for numbers in A001105 (2*n^2).

The dependent definitions are A001783’s product of the integers from one through m that are relatively prime to m, A183300’s positive integers not of the form 2*k^2, and A001105’s integers of the form 2*k^2. The complete quantified reading is

forall n : Nat, 0 < n -> (Odd (a n) <-> not exists k : Nat, n = 2*k^2),

where a(n) sums the phitorials over the positive divisors of n.

Motivation

The independent question is the parity characterization printed as Krizek’s named A280258 conjecture. Issue #8471 registered the verbatim source, full quantifiers, Tier 1 classification, and literature readings before the proof probe.

Gap

The inspected A280258 entry labels the statement Conjecture and contains no settlement marker or linked proof. The A001783, A183300, and A001105 entries supply definitions and the even-divisor component, but do not settle the phitorial divisor-sum statement.

Repository searches on the protected base found no A280258, A001783, A183300, phi-torial, phitorial, or exact-conclusion owner. Pinned Mathlib contains the divisor-cardinality factorization and finite-sum parity machinery, but no phitorial definition or A280258 theorem.

The public theorem has proof_shape: bind-only, escape_witness: none, and admission_basis: open-problem-resolution. It is the module’s sole theorem and settles exactly the preregistered external assertion. There is no direct frozen repository dependency. The proof instantiates pinned Mathlib declarations, including Nat.card_divisors, Nat.coprime_prod_right_iff, Nat.coprime_two_left, Finset.isSquare_prod, Nat.prod_primeFactors_pow_factorization, Nat.factorization_pow, and Finset.odd_sum_iff_odd_card_odd, together with divisor/filter/product lemmas and normalization.

utility: none applies because the definitions and result are unbounded symbolic mathematics, not bounded enumeration, checker infrastructure, numeric reduction, or a certified finite instance.

Route

  1. The phitorial is odd exactly at one and at even inputs. For odd inputs greater than one, the factor two occurs; for even inputs, every totative is odd.
  2. Finite-sum parity reduces Odd (a n) to the parity of the number of divisors having odd phitorial. This set is one together with the even divisors.
  3. If n is odd there are no even divisors. If n=2*m, halving gives a bijection between the even divisors of n and the divisors of m.
  4. The prime-factorization formula makes the number of divisors of positive m odd exactly when m is a square. Hence a(n) is even exactly when n=2*k^2.

Falsifier

Any positive natural n for which a(n) is odd while n=2*k^2, or a(n) is even while no such k exists, would refute the result. A mismatch between the inclusive totative product and A001783 would also refute the transcription; at m=1 the endpoint contributes one, and at m>1 it is filtered out because it is not coprime to m.

Evidence

The Lean definitions phitorial, a, and claim transcribe the source, and result proves the full positive-natural biconditional. The result is frozen with statement id sha256:8e28f255cec6a96e6582d472c6010bfbc996028a0ae8db32d84cfa6b53b6064a and uses exactly [propext, Classical.choice, Quot.sound].

Triage

theorem; resolution proved for the A280258 parity conjecture. The Scribe theorem node carries the matching OpenProblemResolutionClaim with ResolutionKind.Proved.

ASSUMED-UNVERIFIED

The inspected OEIS surfaces do not establish global historical novelty. A complete prior proof outside the searched repository, pinned Mathlib, and OEIS entry scope remains ASSUMED-UNVERIFIED. No priority claim is made.