Conjectures associated with A105210

The question

conjecture_disjoint_starting_values. Cormier and Selfridge found 5 starting values for which the sequences appear to not merge. The sequences were checked up to 10^8.

conjecture. This suggests that there may be infinitely many different (non-merging) sequences obtained by choosing different starting values.

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«105210» (2 statements).

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 = ∅

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 = ∅)

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• A105210

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.