bibkey: luschny2026a398189 authors: Peter Luschny year: 2026 title: OEIS A398189, binary valuations of generalized Schenker sums doi: null url: https://oeis.org/A398189 claim: “For odd n and positive k, the valuation is zero for odd k and equals v2(k+2) for even k outside the residue class 14 modulo 16.” strata_touched:

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

Binary valuations of truncated Schenker sums

The entry defines its table through A398187: S(n,k) = sum(j=0..n-k, (n-k)!/j! * n^j). The comments label the two positive-k formulas in the claim as conjectures. The class k = 14 modulo 16 is explicitly left without a simple formula.

Verified locator

url: https://oeis.org/A398189

The complete internal entry was fetched and read on 2026-09-10, revision #19 Aug 04 2026 01:11:24. The directly referenced entries A398187, A063170 and A000120 were also fetched and read in full. The COMMENTS of A398189 contain the two formulas and the exceptional-class boundary. The FORMULA of A398187 contains the defining sum and horizontal recurrence.

Published background

T. Amdeberhan, D. Callan and V. Moll, Valuations and combinatorics of truncated exponential sums, Integers 13 (2013), A21: https://math.colgate.edu/~integers/n21/n21.pdf . All 16 pages were read; page 5 was also inspected as a rendered image. Section 2, pages 4–5, proves the k=0 formulas. Lemma 2.2, equations (2.11)–(2.14), gives the strictly larger valuation of every positive-index summand when n is even. Truncating and scaling preserves this strict comparison, so the even-n branch is published background as well. Sections 3–4 concern odd primes for the original k=0 sums; Section 5 gives Abel identities and tree interpretations. None of these supplies the odd-n, positive-k valuation formulas proved by the present module.

Search receipt

Searches on 2026-09-10 preceded the local proof. Repository D5 searches for A398189, A063170, Schenker, and truncated factorial/exponential sums found no target theorem. Public interfaces of the factorial quotient, factorial product parity, and alternating factorial sum candidates did not provide the needed cancellation theorem. The same name search in pinned Mathlib db584cd6d46c92f209a44c0f1c829460d327499d found no target. Mathlib supplies factorial divisibility, modular casts, power periodicity and the divisibility characterization of padicValNat used in the proof.

GitHub code search returned zero results for Schenker language:Lean and A398189 language:Lean. The query "truncated exponential" language:Lean returned kim-em/hex-dev/HexTruncatedSeriesMathlib/Newton.lean; its ofPowerSeries_exp was read and concerns truncation of formal power series, without this integer valuation statement. The OEIS entry and its direct references above did not report a proof of the remaining formulas. The result is suspected novel within this searched scope, not a claim of exhaustive literature coverage. Further references linked by the three direct OEIS entries were not opened and remain ASSUMED-UNVERIFIED.

Formal scope

The local proof identifies the natural-number sum with its recurrence, expands six steps modulo 16 and removes the old history using the product of six consecutive integers. Odd powers have period four modulo 16. Private residue lemmas and the separate short-length cases are lifted to all n and k, then nonzero residues give exact binary valuations below four. The public theorem requires odd n and 1 <= k <= n and gives both requested branches. It makes no claim about the exceptional class.