bibkey: wiseman2025a383093
authors: Gus Wiseman
year: 2025
title: A383093 — partitions admitting constant blocks with a common sum
doi: null
url: https://oeis.org/A383093
claim: The entry conjectures that the divisor sum of capable partition counts equals A323774.
strata_touched:
  - D5/S1/Words/Compositions/ConstantEqualSumDivisorIdentity
license: citation-only
triage: anchor

OEIS A383093

Verified locator

url: https://oeis.org/A383093

doi: null

Revision 13 of https://oeis.org/A383093/internal is dated 2025-05-04; the author date is 2025-04-22. Its formula states:

Conjecture: We have Sum_{d|n} a(d) = A323774(n), so this is the Moebius transform of A323774.

Revision 19 of https://oeis.org/A323774/internal, dated 2024-11-30, counts unlabeled multiset partitions into constant blocks with equal sums. Its binomial divisor formula counts all systems; it does not prove the conjectured transform of the existence count.

Definitions and proof

A block with positive common sum D and positive value x dividing D contains D/x copies of x. A multiset b of such values records every block, including repeated identical blocks. Its total weight is D times the cardinality of b. At weight zero there is one empty system.

The existence count uses Mathlib integer partitions and asks whether some D and b flatten to that partition. The system count independently counts pairs (D,b) of the required weight. Neither definition uses a divisor sum. The Lean proof establishes the multiplicity criterion and the unique reconstruction of b from the flattened partition and D.

Let L be the lcm of the support. Every admissible D is tL. Dividing all multiplicities by t gives a capable partition with the same support, canonical common sum L, and weight n/t. Conversely, multiplying a capable partition’s multiplicities by n/d gives exactly one system at weight n. Equality of support lcms recovers the scaling factor and proves uniqueness.

Proof scope

Mathlib supplies the lcm API and Multiset.exists_smul_of_dvd_count, which is used directly. ConstantBlocksDistinctRunSums concerns distinct sums and does not supply the equal-sum normalization.

The main entry’s normal-system cross-reference A382203 has distinct sums; A382204 is the corresponding equal-sum entry. This discrepancy does not alter either target definition.

The Lean construction is repository work. No publication priority or exhaustive literature coverage is claimed.