243106.lean`

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

7 declarations, 1 contain sorry, 0 declared as axioms

Signed digit sums in bases $b \ge 5$

The sequence is defined by $$a(n) = \sum_{k=1}^n (-1)^{\operatorname{isprime}(k)} 10^k$$ where the sign is $-1$ if $k$ is prime, and $1$ if $k$ is not prime.

Generalizing the digit pattern observed in $a(n)$, it was conjectured on the OEIS entry that for any base $b \ge 5$ and any choice of signs $\pm 1$, the absolute value of $\sum_{k=1}^n \pm b^k$ only contains digits in $\{0, 1, b-2, b-1\}$.

References:

A243106
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

Imports (1)

FormalConjecturesUtil

def OeisA243106.a The sequence $$a(n) = \sum_{k=1}^n (-1)^{\operatorname{isprime}(k)} 10^k$$ where the sign is $-1$ if $k$ is prime, and $1$ if $k$ is not prime line 48 source
Source documentation
The sequence $$a(n) = \sum_{k=1}^n (-1)^{\operatorname{isprime}(k)} 10^k$$ where the sign is $-1$ if $k$ is prime, and $1$ if $k$ is not prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Int
Proof uses: Finset.sum, Finset.Icc, OfNat.ofNat, HMul.hMul, ite, Nat.Prime, Neg.neg, HPow.hPow
13 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : Int :=
…
```
lemma OeisA243106.a_1 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA243106.a, OfNat.ofNat, Nat
Proof uses: Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA243106.a_2 line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA243106.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA243106.a_3 line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA243106.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA243106.a_4 line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA243106.a, OfNat.ofNat, Nat
Proof uses: Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma OeisA243106.a_5 line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Int, OeisA243106.a, OfNat.ofNat, Nat, Neg.neg
Proof uses: Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA243106.digits_sum_restricted sorry line 77 source
depends on sorry what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Int, Membership.mem, Finset, Finset.Icc, Or, Eq, Neg.neg, Finset.sum, HMul.hMul, HPow.hPow, List, Nat.digits, Int.natAbs, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.