117531.lean`

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

7 declarations, 1 contain sorry, 0 declared as axioms

Number of primes in $n$-th row of triangle $k^2 - k + p_n$

$a(n)$ is the number of primes in the $n$-th row of the triangle $T(n, k) = k^2 - k + p_n$ for $1 \le k \le n$, where $p_n$ is the $n$-th prime ($p_1=2, p_2=3, \dots$).

References:

A117531

Imports (1)

FormalConjecturesUtil

def OeisA117531.a Number of primes in the $n$-th row of $T(n, k) = k^2 - k + p_n$ for $1 \le k \le n$ line 34 source
Source documentation
Number of primes in the $n$-th row of $T(n, k) = k^2 - k + p_n$ for $1 \le k \le n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.nth, Nat.Prime, HSub.hSub, OfNat.ofNat, Finset.card, Finset.filter, HAdd.hAdd, HPow.hPow, Finset.Icc
12 instances, coercions and auxiliary constants left out.
```
noncomputable def a (n : ℕ) : ℕ :=
…
```
theorem OeisA117531.a_0 Value of the sequence a at 0 line 40 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA117531.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 0 := by rfl
…
```
Mentions: OeisA117531.a
theorem OeisA117531.a_1 Value of the sequence a at 1 line 44 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA117531.a, OfNat.ofNat
Proof uses: id, Nat.nth, Nat.Prime, HSub.hSub, Nat.nth_prime_zero_eq_two, Finset.card, Finset.filter, HAdd.hAdd, HPow.hPow, Finset.Icc, Eq.mpr, congrArg, Finset, Singleton.singleton, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl, Finset.ext and 20 more
26 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by
…
```
Mentions: OeisA117531.a
theorem OeisA117531.a_2 Value of the sequence a at 2 line 61 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA117531.a, OfNat.ofNat
Proof uses: id, Nat.nth, Nat.Prime, HSub.hSub, Nat.nth_prime_one_eq_three, Finset.card, Finset.filter, HAdd.hAdd, HPow.hPow, Finset.Icc, Eq.mpr, congrArg, Finset, Insert.insert, Singleton.singleton, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl and 29 more
30 instances, coercions and auxiliary constants left out.

Tactic trace: 24 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 2 := by
…
```
Mentions: OeisA117531.a
theorem OeisA117531.a_3 Value of the sequence a at 3 line 80 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA117531.a, OfNat.ofNat
Proof uses: id, Nat.nth, Nat.Prime, HSub.hSub, Nat.nth_prime_two_eq_five, Finset.card, Finset.filter, HAdd.hAdd, HPow.hPow, Finset.Icc, Eq.mpr, congrArg, Finset, Insert.insert, Singleton.singleton, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl and 28 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
```
theorem a_3 : a 3 = 3 := by
…
```
Mentions: OeisA117531.a
theorem OeisA117531.a_4 Value of the sequence a at 4 line 101 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA117531.a, OfNat.ofNat
Proof uses: id, Nat.nth, Nat.Prime, HSub.hSub, Nat.nth_prime_three_eq_seven, Finset.card, Finset.filter, HAdd.hAdd, HPow.hPow, Finset.Icc, Eq.mpr, congrArg, Finset, Insert.insert, Singleton.singleton, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl and 38 more
31 instances, coercions and auxiliary constants left out.

Tactic trace: 36 steps
Open to load the checked before-and-after goal states.
```
theorem a_4 : a 4 = 3 := by
…
```
Mentions: OeisA117531.a
theorem OeisA117531.conjecture Conjecture: $a(n) < n$ for $n > 13$ sorry line 127 source
Source documentation
Conjecture: $a(n) < n$ for $n > 13$.

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, LT.lt, OeisA117531.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) (h : n > 13) : a n < n := by
…
```
Mentions: OeisA117531.a