105210.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

12 declarations, 2 contain sorry, 0 declared as axioms

Conjectures associated with A105210

$a(1) = 393$; for $n > 1$, $a(n) = a(n-1)$ + 1 + sum of distinct prime factors of $a(n-1)$ that are $< a(n-1)$.

References:

A105210

Imports (1)

FormalConjecturesUtil

def OeisA105210.sumRelevantPrimeFactors The sum of distinct prime factors of $n$ that are strictly less than $n$ line 37 source
Source documentation
The sum of distinct prime factors of $n$ that are strictly less than $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Nat.primeFactors, ite, LT.lt, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
```
def sumRelevantPrimeFactors (n : ℕ) : ℕ :=
…
```
def OeisA105210.a The primary defining sequence a line 43 source
Source documentation
The primary defining sequence a. $a(1) = 393$; for $n > 1$, $a(n)$ = $a(n-1)$ + 1 + sum of distinct prime factors of $a(n-1)$ that are < $a(n-1)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: WellFounded.Nat.fix, InvImage, LT.lt, OeisA105210.a.match_1, Unit, OfNat.ofNat, Nat.succ, HAdd.hAdd, OeisA105210.sumRelevantPrimeFactors
5 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℕ
| 0 => 0
| 1 => 393
| n + 2 =>
  let aPrev := a (n + 1)
…
```
theorem OeisA105210.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 53 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

extra axioms: _private.0.OeisA105210.a_1._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA105210.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 393 := by native_decide
…
```
Mentions: OeisA105210.a
theorem OeisA105210.a_2 line 56 source
extra axioms: _private.0.OeisA105210.a_2._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA105210.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA105210.a_3 line 59 source
extra axioms: _private.0.OeisA105210.a_3._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA105210.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA105210.a_4 line 62 source
extra axioms: _private.0.OeisA105210.a_4._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA105210.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA105210.a_5 line 65 source
extra axioms: _private.0.OeisA105210.a_5._native.native_decide.ax_1 what this means
Statement uses: Eq, Nat, OeisA105210.a, OfNat.ofNat
Proof uses: of_decide_eq_true
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def OeisA105210.nextTermFn The transition function for these sequences: $f(n) = n + 1 + S(n)$ line 68 source
Source documentation
The transition function for these sequences: $f(n) = n + 1 + S(n)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HAdd.hAdd, OfNat.ofNat, OeisA105210.sumRelevantPrimeFactors
3 instances, coercions and auxiliary constants left out.
```
def nextTermFn (n : ℕ) : ℕ :=
…
```
def OeisA105210.sequenceStartingAt The sequence starting at positive integer $k$, where seq $k$ $n$ is the $n$-th term (1-indexed) line 73 source
Source documentation
The sequence starting at positive integer $k$, where seq $k$ $n$ is the $n$-th term (1-indexed). This is the general form of the sequence A105210 for a starting value $k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: WellFounded.Nat.fix, InvImage, LT.lt, OeisA105210.sequenceStartingAt.match_1, Unit, OfNat.ofNat, Nat.succ, OeisA105210.nextTermFn
3 instances, coercions and auxiliary constants left out.
```
def sequenceStartingAt (k : ℕ) : ℕ → ℕ
| 0 => 0
| 1 => k
| n + 1 => nextTermFn (sequenceStartingAt k n)
termination_by n => n

/-- The set of all terms generated by the sequence starting at $k$. -/
```
Mentions: OeisA105210.nextTermFn
def OeisA105210.sequenceSet The set of all terms generated by the sequence starting at $k$ line 80 source
Source documentation
The set of all terms generated by the sequence starting at $k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, Exists, And, LE.le, OfNat.ofNat, Eq, OeisA105210.sequenceStartingAt
2 instances, coercions and auxiliary constants left out.
```
def sequenceSet (k : ℕ) : Set ℕ :=
…
```
theorem OeisA105210.conjecture_disjoint_starting_values Cormier and Selfridge found 5 starting values for which the sequences appear to not merge sorry line 88 source
Source documentation
Cormier and Selfridge found 5 starting values for which the sequences appear to not merge. The sequences were checked up to 10^8.

depends on sorry what this means
Statement uses: Nat, Membership.mem, Set, Insert.insert, OfNat.ofNat, Singleton.singleton, Ne, Eq, Inter.inter, OeisA105210.sequenceSet, EmptyCollection.emptyCollection
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture_disjoint_starting_values :
    ∀ j k : ℕ,
      j ∈ ({1, 393, 412, 668, 932} : Set ℕ) →
      k ∈ ({1, 393, 412, 668, 932} : Set ℕ) →
      j ≠ k →
      sequenceSet j ∩ sequenceSet k = ∅ := by
…
```
Mentions: OeisA105210.sequenceSet
theorem OeisA105210.conjecture This suggests that there may be infinitely many different (non-merging) sequences obtained by choosing different starting values sorry line 101 source
Source documentation
This suggests that there may be infinitely many different (non-merging) sequences obtained by choosing different starting values.

depends on sorry what this means
Statement uses: Exists, Set, Nat, And, Set.Infinite, Membership.mem, LE.le, OfNat.ofNat, Ne, Eq, Inter.inter, OeisA105210.sequenceSet, EmptyCollection.emptyCollection
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture :
  ∃ K : Set ℕ,
    Set.Infinite K ∧
    (∀ k ∈ K, 1 ≤ k) ∧
    (∀ j k : ℕ,
      j ∈ K → k ∈ K → j ≠ k →
      sequenceSet j ∩ sequenceSet k = ∅) := by
…
```
Mentions: OeisA105210.sequenceSet