Hanna’s A363569 Prime-Index Formula

Abstract

Hanna’s A363569 prime-index coefficient formula.

Definition 1.1 (Hanna’s A363569 conjecture).

$$claim=(\forall p:\mathbb{N},(\mathrm{Nat}.\mathrm{Prime}\left(p\right))⟹((3<p)⟹(OeisA363569PowerOfTwo.a\left(p\right)=(p:\mathbb{C})\cdot ((-1){)}^{\mathrm{Nat}.\mathrm{div}\left(p-1,2\right)}-1)))$$

Formalization. D5/S1/Recurrence/HannaBilateral/OeisA363569Prime.claim (✓ std3).

Citation. Paul D. Hanna (2023). OEIS A363558, A363559, A363561 and A363569: bilateral power sums. URL: https://oeis.org/A363558.

Commentary.

a(p) = p*(-1)^((p-1)/2) - 1 for primes p > 3 (conjecture). Source: OEIS A363569, FORMULA field (an unpaginated entry). Indices are natural numbers; values and numeric expressions on both sides are complex. Nat.div denotes natural-number division, so division by two is integer division.

Theorem 1.2 (The A363569 prime-index formula).

$$\forall p:\mathbb{N},(\mathrm{Nat}.\mathrm{Prime}\left(p\right))⟹((3<p)⟹(OeisA363569PowerOfTwo.a\left(p\right)=(p:\mathbb{C})\cdot ((-1){)}^{\mathrm{Nat}.\mathrm{div}\left(p-1,2\right)}-1))$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/HannaBilateral/OeisA363569Prime.result (✓ std3). ∎

Resolves. Problems/oeis-a363569-hanna-bilateral-prime-values (proved) by D5/S1/Recurrence/HannaBilateral/OeisA363569Prime.result.

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2023). OEIS A363558, A363559, A363561 and A363569: bilateral power sums. URL: https://oeis.org/A363558.

Commentary.

Apply the complex-parameter prime coefficient identity and simplify the indicated parameter powers.

References

• Truth anchor: D5/S1/Recurrence/HannaBilateral/OeisA363569Prime.claim
• Truth anchor: D5/S1/Recurrence/HannaBilateral/OeisA363569Prime.result
• Dependency: D5/S1/Recurrence/HannaBilateral/OeisA363558PowerOfTwo
• Dependency: D5/S1/Recurrence/HannaBilateral/OeisA363569PowerOfTwo