110475.lean`

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

8 declarations, 1 contain sorry, 0 declared as axioms

Number of symbols '*' and '^' to write the canonical prime factorization of n

The canonical prime factorization is $n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$. The written form is $p_1^{\wedge} e_1 * p_2^{\wedge} e_2 * \cdots * p_k^{\wedge} e_k$, where the $\wedge$ appears only if $e_i > 1$. $a(n) = (\text{number of distinct prime factors}) - 1 +$ $(\text{number of distinct prime factors with exponent } > 1)$.

References:

A110475

Imports (1)

FormalConjecturesUtil

def OeisA110475.a The primary defining sequence a line 41 source
Source documentation
The primary defining sequence a. $a(n)$ is the number of symbols '*' and '^' to write the canonical prime factorization of $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finsupp, Nat.factorization, Finset, Finsupp.support, Finset.card, HSub.hSub, OfNat.ofNat, Finset.filter, GT.gt, HAdd.hAdd
11 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA110475.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 51 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA110475.a, OfNat.ofNat
Proof uses: id, of_eq_true, HAdd.hAdd, Finset.card, Finset.filter, GT.gt, Finsupp, Nat.factorization, EmptyCollection.emptyCollection, Finset, Eq.trans, True, congrFun', congrArg, False, Finset.filter_congr, Membership.mem, Iff.of_eq, LT.lt, Nat.factorization_one and 3 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 0 := by
…
```
Mentions: OeisA110475.a
theorem OeisA110475.a_2 line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA110475.a, OfNat.ofNat
Proof uses: id, Nat.Prime, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, HAdd.hAdd, HSub.hSub, Finset.card, Insert.insert, Finset, EmptyCollection.emptyCollection, Finset.filter, GT.gt, Finsupp, Nat.factorization, Finsupp.single, congrArg and 21 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA110475.a_3 line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA110475.a, OfNat.ofNat
Proof uses: id, Nat.Prime, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, HAdd.hAdd, HSub.hSub, Finset.card, Insert.insert, Finset, EmptyCollection.emptyCollection, Finset.filter, GT.gt, Finsupp, Nat.factorization, Finsupp.single, congrArg and 21 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem OeisA110475.a_4 line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA110475.a, OfNat.ofNat
Proof uses: id, HPow.hPow, Eq.refl, Eq.mpr, HAdd.hAdd, HSub.hSub, Finset.card, Insert.insert, Finset, EmptyCollection.emptyCollection, Finset.filter, GT.gt, Finsupp, Nat.factorization, congrArg, Finsupp.single, Nat.Prime.factorization_pow, of_decide_eq_true, Nat.Prime, Bool and 23 more
51 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
theorem OeisA110475.a_5 line 77 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA110475.a, OfNat.ofNat
Proof uses: id, Nat.Prime, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Eq.mpr, HAdd.hAdd, HSub.hSub, Finset.card, Insert.insert, Finset, EmptyCollection.emptyCollection, Finset.filter, GT.gt, Finsupp, Nat.factorization, Finsupp.single, congrArg and 21 more
44 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
def OeisA110475.exceptionalSet The set of exceptional integers line 86 source
Source documentation
The set of exceptional integers.

standard axioms: Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: Insert.insert, OfNat.ofNat, Singleton.singleton
4 instances, coercions and auxiliary constants left out.
```
def exceptionalSet : Finset ℕ :=
…
```
theorem OeisA110475.conjecture It is conjectured that $1,2,3,4,5,6,7,9,11$ are the only positive integers which cannot be represented as the sum of two elements of indices $n$ such that $a(n) = 1$ sorry line 94 source
Source documentation
It is conjectured that $1,2,3,4,5,6,7,9,11$ are the only positive integers which cannot be represented as the sum of two elements of indices $n$ such that $a(n) = 1$.

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Iff, Not, Membership.mem, Finset, OeisA110475.exceptionalSet, Exists, And, Eq, OeisA110475.a, HAdd.hAdd
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 :
  ∀ m > 0, m ∉ exceptionalSet ↔ ∃ x y : ℕ, a x = 1 ∧ a y = 1 ∧ m = x + y := by
…
```
Mentions: OeisA110475.exceptionalSet, OeisA110475.a