Representations as p + 2^x + 11 · 2^y with p ≡ 1 pmod 6

On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n) = 0 if and only if n < 16 or n ∈ 18, 21, 24, 51, 84, 1011, 59586; in other words, except for 35, 41, 47, 101, 167, 2021, 119171, any odd integer greater than 30 can be written as the sum of a prime congruent to 1 bmod 6, a positive power of 2 and…

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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-157237,
  title        = {Representations as p + 2^x + 11 · 2^y with p ≡ 1 pmod 6},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-157237}},
  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

On Feb. 24, 2009, Zhi-Wei Sun conjectured that $a (n) = 0$ if and only if $n < 16$ or $n \in {18, 21, 24, 51, 84, 1011, 59586}$ ; in other words, except for $35, 41, 47, 101, 167, 2021, 119171$ , any odd integer greater than $30$ can be written as the sum of a prime congruent to $1 mod 6$ , a positive power of $2$ and eleven times a positive power of $2$ .

Number of ways to write the $n$ -th positive odd integer in the form $p + 2^{x} + 11 \cdot 2^{y}$ with $p$ a prime congruent to $1 mod 6$ and $x, y$ positive integers. $a (n) = {(p, x, y) : p + 2^{x} + 11 \cdot 2^{y} = 2 n - 1 with p prime, p \equiv 1 (mod 6), x, y \in Z^{+}} .$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«157237».

theorem conjecture (n : ℕ) (hn : 0 < n) :
    a n = 0 ↔ n ≤ 15 ∨ n = 18 ∨ n = 21 ∨ n = 24 ∨ n = 51 ∨ n = 84 ∨ n = 1011 ∨ n = 59586

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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

A157237
Z.-W. Sun, "Mixed sums of primes and other terms", arXiv preprint arXiv:0901.3075 [math.NT], 2009.

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.