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import FormalConjecturesUtilSmallest prime formed by concatenation $n, n-1, \dots, n-k$
$a(n)$ is the smallest prime which has the form of the concatenation $n, n-1, n-2, \dots, n-k$ for some $k < n$, or $0$ if no such prime exists.
References:
namespace OeisA87571Concatenate a list of numbers into a list of decimal digits in most-significant-first order.
def getAllDigitsMsf (L : List ℕ) : List ℕ :=
List.flatten (L.map fun k => (Nat.digits 10 k).reverse)Convert a list of decimal digits (most significant first) to a natural number.
def ofDigitsMsf (D : List ℕ) : ℕ :=
D.foldl (fun acc d => acc * 10 + d) 0Concatenation of numbers $n, n-1, \dots, n-k$.
def concatNum (n : ℕ) (k : ℕ) : ℕ :=
ofDigitsMsf (getAllDigitsMsf ((List.range (k + 1)).map fun i => n - i))Smallest prime of the form $n, n-1, \dots, n-k$ for $k < n$, or $0$ if none exists.
def a (n : ℕ) : ℕ :=
match ((List.range n).map (concatNum n)).find? Nat.Prime with
| some p => p
| none => 0
Value of the sequence a at 0.
@[category test, AMS 11]
theorem a_0 : a 0 = 0 := ⊢ a 0 = 0 All goals completed! 🐙
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 0 := ⊢ a 1 = 0 All goals completed! 🐙
Value of the sequence a at 2.
@[category test, AMS 11]
theorem a_2 : a 2 = 2 := ⊢ a 2 = 2 All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 3 := ⊢ a 3 = 3 All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 43 := ⊢ a 4 = 43 All goals completed! 🐙Conjecture: There are infinitely many composite numbers $n$ such that $a(n)$ is nonzero.
@[category research open, AMS 11]
theorem conjecture (M : ℕ) : ∃ n > M, 1 < n ∧ ¬ n.Prime ∧ a n ≠ 0 := M:ℕ⊢ ∃ n > M, 1 < n ∧ ¬Nat.Prime n ∧ a n ≠ 0
All goals completed! 🐙end OeisA87571