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import FormalConjecturesUtilDenominator of $\sum_{k=1}^n k^{\mu(k)}$
The sequence $a(n)$ is the denominator of $\sum_{k=1}^n k^{\mu(k)}$, where $\mu$ is the Möbius function.
References:
namespace OeisA80326open ArithmeticFunction$k^{\mu(k)}$ as a rational number.
def term (k : ℕ) : ℚ :=
let mu := (moebius k : ℤ)
if mu = 1 then k
else if mu = 0 then 1
else (1 : ℚ) / kDenominator of $\sum_{k=1}^n k^{\mu(k)}$.
def a (n : ℕ) : ℕ :=
(∑ k ∈ Finset.Icc 1 n, term k).den
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1
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 = 6 := ⊢ a 3 = 6
All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 6 := ⊢ a 4 = 6
All goals completed! 🐙
Value of the sequence a at 5.
@[category test, AMS 11]
theorem a_5 : a 5 = 30 := ⊢ a 5 = 30
All goals completed! 🐙Conjecture: $a(n) = \text{primorial}(n)$ for infinitely many $n$.
@[category research open, AMS 11]
theorem conjecture : {n : ℕ | a n = primorial n}.Infinite := ⊢ {n | a n = primorial n}.Infinite
All goals completed! 🐙end OeisA80326