Totient Periods for Two Divisor-Factorial Sequences

Abstract

Divisor-factorial tails prove Bala’s A258899 and A258903 totient-period conjectures.

The source is the COMMENTS and FORMULA fields of OEIS A258899 and A258903, which are not paginated. Both sequences have offset zero and value 1 at index zero. The modulo-7 example in the A258903 comment begins at index 1. All indices, exponents, factorials and sums here are natural numbers. Nat.div denotes natural-number division, including the factorial quotient; it is not rational division.

Definition 1.1 (The sequence A258899).

$$\forall n:\mathbb{N},a258899\left(n\right)=\mathrm{if}n=0\mathrm{then}1\mathrm{else}\sum _{d\in Nat.divisors\left(n\right)}(2^d\cdot Nat.div\left(Nat.factorial\left(n\right),Nat.factorial\left(d\right)\cdot Nat.factorial\left(Nat.div\left(n,d\right)\right)\right))$$

Formalization. D5/S1/Recurrence/Periodic/BalaDivisorFactorialTotientPeriod.a258899 (✓ std3).

Citation. Paul D. Hanna; Peter Bala (2025). OEIS A258899 and A258903: divisor-factorial sequences and totient-period conjectures. URL: https://oeis.org/A258899.

Commentary.

OEIS A258899, FORMULA: “For n >= 1, a(n) = Sum_{d divides n} 2^d * n!/(d!*(n/d)!).” The initial value is 1, as in the entry’s term list and offset. The sum ranges over Nat.divisors n.

Definition 1.2 (The sequence A258903).

$$\forall n:\mathbb{N},a258903\left(n\right)=\mathrm{if}n=0\mathrm{then}1\mathrm{else}\sum _{d\in Nat.divisors\left(n\right)}(2^d\cdot 3^{Nat.div\left(n,d\right)}\cdot Nat.div\left(Nat.factorial\left(n\right),Nat.factorial\left(d\right)\cdot Nat.factorial\left(Nat.div\left(n,d\right)\right)\right))$$

Formalization. D5/S1/Recurrence/Periodic/BalaDivisorFactorialTotientPeriod.a258903 (✓ std3).

Citation. Paul D. Hanna; Peter Bala (2025). OEIS A258899 and A258903: divisor-factorial sequences and totient-period conjectures. URL: https://oeis.org/A258899.

Commentary.

OEIS A258903, FORMULA: “For n >= 1, a(n) = Sum_{d divides n} 2^d * 3^(n/d) * n!/(d!*(n/d)!).” The initial value is 1, as in the entry’s term list and offset. The sum ranges over Nat.divisors n.

Definition 1.3 (Bala’s two conjectures).

$$(\mathrm{claim})\iff ((\forall m:\mathbb{N},(0<m)\Rightarrow (\exists N:\mathbb{N},\exists T:\mathbb{N},(0<T)\wedge ((T\mid Nat.totient\left(m\right))\wedge (\forall n:\mathbb{N},(N\le n)\Rightarrow (Nat.ModEq\left(m,a258899\left(n+T\right),a258899\left(n\right)\right))))))\wedge (\forall m:\mathbb{N},(0<m)\Rightarrow (\exists N:\mathbb{N},\exists T:\mathbb{N},(0<T)\wedge ((T\mid Nat.totient\left(m\right))\wedge (\forall n:\mathbb{N},(N\le n)\Rightarrow (Nat.ModEq\left(m,a258903\left(n+T\right),a258903\left(n\right)\right)))))))$$

Formalization. D5/S1/Recurrence/Periodic/BalaDivisorFactorialTotientPeriod.claim (✓ std3).

Citation. Paul D. Hanna; Peter Bala (2025). OEIS A258899 and A258903: divisor-factorial sequences and totient-period conjectures. URL: https://oeis.org/A258899.

Commentary.

The COMMENTS field of each entry states: “Conjecture: the sequence a(n) taken modulo a positive integer k is eventually periodic with the period dividing phi(k).” The modulus k is written m. Each conjunct has its own natural onset N and positive natural period T dividing Nat.totient m. Nat.ModEq m compares the two natural terms modulo m. Modulus 1 is included.

Theorem 1.4 (Both totient-period assertions).

$$(\forall m:\mathbb{N},(0<m)\Rightarrow (\exists N:\mathbb{N},\exists T:\mathbb{N},(0<T)\wedge ((T\mid Nat.totient\left(m\right))\wedge (\forall n:\mathbb{N},(N\le n)\Rightarrow (Nat.ModEq\left(m,a258899\left(n+T\right),a258899\left(n\right)\right))))))\wedge (\forall m:\mathbb{N},(0<m)\Rightarrow (\exists N:\mathbb{N},\exists T:\mathbb{N},(0<T)\wedge ((T\mid Nat.totient\left(m\right))\wedge (\forall n:\mathbb{N},(N\le n)\Rightarrow (Nat.ModEq\left(m,a258903\left(n+T\right),a258903\left(n\right)\right))))))$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Periodic/BalaDivisorFactorialTotientPeriod.result (✓ std3). ∎

Resolves. Problems/bala-a258899-totient-period (proved) by D5/S1/Recurrence/Periodic/BalaDivisorFactorialTotientPeriod.result.

Source. Repository-derived.

Acknowledgement. Paul D. Hanna; Peter Bala (2025). OEIS A258899 and A258903: divisor-factorial sequences and totient-period conjectures. URL: https://oeis.org/A258899.

Commentary.

For n at least 2m+4 and a proper factorization n=de, the inequality 2*(d+e) <= n+4 gives d+e+m <= n. Factorial divisibility then makes m divide n!/(d!e!). Thus only the factors 1 and n survive in the sum modulo m, giving uv^n+vu^n for arbitrary natural weights u,v, including zero. The theorem D5.S1.Recurrence.Periodic.StirlingTransformTotientPeriod.pow_add_totient makes both powers unchanged by a shift of Nat.totient m. Weights (2,1) and (2,3) give the two sequences. In both conjuncts N=2m+4 and T=Nat.totient m suffice. Neither is asserted minimal.

References

• Truth anchor: D5/S1/Recurrence/Periodic/BalaDivisorFactorialTotientPeriod.a258899
• Truth anchor: D5/S1/Recurrence/Periodic/BalaDivisorFactorialTotientPeriod.a258903
• Truth anchor: D5/S1/Recurrence/Periodic/BalaDivisorFactorialTotientPeriod.claim
• Truth anchor: D5/S1/Recurrence/Periodic/BalaDivisorFactorialTotientPeriod.result
• Dependency: D5/S1/Recurrence/Periodic/StirlingTransformTotientPeriod