Sum of $a(k)/k!$ over divisors equals harmonic number

formal-conjectures · oeis · ams-11

The sequence $a(n)$ satisfies $\sum_{k \mid n} \frac{a(k)}{k!} = \sum_{j=1}^n \frac{1}{j} = H_n$,
where the sum on the left is over positive divisors $k$ of $n$. By Möbius inversion,
$$a(n) = n! \sum_{d \mid n} \mu(n/d) H_d$$
where $H_d = \sum_{j=1}^d \frac{1}{j}$ is the $d$-th harmonic number.

The terms are not all positive. The first negative one is
$a(30) = -22690644647302814715858124800000$.
Conjecture: $a(n) < 0$ if and only if A001221(n) is an odd number $\ge 3$.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A067857
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/67857.lean
Local target: Corpus.OEIS67857.conjecture
Source SHA-256: 94594d0b2630d031ac057a0892913fe2878de4a87b1bec315e8a04c9258e72e7
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

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