# Permutations with no whole-number averages

Partial progress on an open problem. Vamshi Jandhyala.

> If no proper block of a permutation of 1..n averages to a whole number, then n = 2ᵐ − 1; the asker's example works exactly when n is prime, and n = 15, 63 have none.

Canonical: https://vamshij.com/research/good-permutations-mersenne
Code and Lean proofs: https://github.com/jvvk/mathematics/tree/main/good-permutations-mersenne
Paper (PDF): https://github.com/jvvk/mathematics/blob/main/good-permutations-mersenne/paper/note.pdf

Call a permutation of `1, …, n` *good* if no block of consecutive entries, of length at least 2 and other than the whole permutation, has an integer average. Philip Weiss asked on MathOverflow ([question 514690](https://mathoverflow.net/q/514690)) whether, for odd `n`, good permutations exist exactly when `n` is a Mersenne prime. Sergiu Alexandru Bîsceanu's [answer](https://mathoverflow.net/a/514702), building on a comment by te4, shows that `n` must be `2^m − 1` (Theorem 1). Whether a composite `2^m − 1` can have a good permutation is open.

This note adds:

- **Theorem 2.** For `n = 2^m − 1`, the asker's permutation `1, n−1, n, n−3, n−2, …, 2, 3` has an integer-average block exactly at the prefixes whose length divides `n`. So it is good if and only if `n` is prime. The proof reads off every block average from the closed form `a_t = n + 2 − t − (−1)^t`.
- **Theorem 3.** For `n = 7` and `31` there are exactly four good permutations (the asker's, its reverse, its complement and the reverse of the complement); for the composite `n = 15` and `63` there are none. The search for `63` uses the structure in Bîsceanu's proof (Corollary 5: `a_q = q` and positions `i`, `i + q` hold values `q` apart), which halves the search.

Preprint v1, 10 October 2026, not peer reviewed. Theorem 1 (with te4's congruence), Corollary 5 and Theorem 2 are formally verified in Lean 4 (`lean/`, standard axioms only); Theorem 3 is computational. The author used AI tools (Claude, Anthropic) in this work, as described in the paper's acknowledgement, and is responsible for its content.