Ascending descending base exponent transform of squares

The smallest prime in this sequence is a(2) = 5. What is the next prime?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-oeis-113257,
  title        = {Ascending descending base exponent transform of squares},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-113257}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

conjecture1. The smallest prime in this sequence is $a (2) = 5$ . What is the next prime?

conjecture2. What is the first square value after 1?

a n is $\sum_{i = 1}^{n} (i^{2})^{(} (n - i + 1)^{2})$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«113257» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem conjecture1 :
    answer(sorry) = a (sInf {n : ℕ | 2 < n ∧ (a n).Prime})

theorem conjecture2 :
    answer(sorry) = a (sInf {n : ℕ | 1 < n ∧ IsSquare (a n)})

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A113257

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.