slug: oeis-a339621-fibonacci-divisor-sum-even-index bibkey: lagneau2020a339621 doi: null url: https://oeis.org/A339621 triage: theorem motivation_gids:

• D5/S1/Scale/Fibonacci

Fibonacci divisor sums at square-plus-one arguments have even index

Problem

OEIS A339621 records these lines verbatim:

%N Sum of Fibonacci divisors of n^2 + 1.
%C A Fibonacci divisor of a number k is a Fibonacci number that divides k. (The divisor 1 is only counted once.)
%C Conjecture: If the sum of the Fibonacci divisors of m^2 + 1 is a Fibonacci number, then this number belongs to the sequence A001906(n) = F(2n) where F(n) is the Fibonacci sequence.
%A _Michel Lagneau_, Dec 10 2020

Formally, let FibValues(N) be the finite set of distinct positive Fibonacci values dividing N, so the value 1 occurs once. Every positive Fibonacci value has a unique canonical index r >= 2. For every natural m, if sum(FibValues(m^2 + 1)) is a Fibonacci value and r >= 2 is its canonical index, then there is a positive integer q such that sum(FibValues(m^2 + 1)) = F_(2q).

Motivation

The conjecture asks whether a divisibility-defined set can sum to a Fibonacci value with odd canonical index r >= 3, equivalently one with no even-index representation. Carlitz’s classification answers the stronger question for every finite set of distinct Fibonacci values that contains 1, without using the square-plus-one form.

Gap

The OEIS comment labels the statement a conjecture, but Leonard Carlitz’s 1968 Theorem 2 already classifies every representation of a Fibonacci number as a sum of distinct Fibonacci values. The A339621 statement is therefore literature-attested as an immediate corollary rather than an open target.

Route

Carlitz proves R(F_r) = floor(r/2) and lists all representations as S_(r,k) = {F_(r-2k)} union {F_(r-2j-1) : 0 <= j < k}. The A339621 divisor set is finite, has distinct Fibonacci values, and contains 1. If its sum is a Fibonacci value, choose its unique canonical index r >= 2; the divisor set is then one of Carlitz’s sets S_(r,k). Among those classified representations, one that contains the value 1 can occur only when r is even. Hence an even-index representation F_r = F_(2q) exists for some positive q, which is precisely membership in A001906. No property special to m^2 + 1 is needed. For m = 0, the sum is 1; its canonical index is 2 even though the same value also equals F_1.

Falsifier

A counterexample would be a natural m whose distinct Fibonacci divisors of m^2 + 1, counting 1 once, sum to a Fibonacci value that has no even-index representation. Equivalently, the value’s canonical index r >= 2 would be odd, and hence r >= 3. Such a set would contradict Carlitz’s complete representation classification.

Evidence

• OEIS source: Library/notes/lagneau2020a339621.md.
• Published classification: Leonard Carlitz, “Fibonacci Representations”, The Fibonacci Quarterly 6(4) (1968), 193-220, Theorem 2, DOI 10.1080/00150517.1968.12431213.
• Sam Chow and Tom Slattery, “On Fibonacci partitions”, Journal of Number Theory 225 (2021), 310-326, equation (1.1), reproduces Carlitz’s count.

Triage

theorem, literature-attested. The deduction above resolves the OEIS statement at the literature layer. This dossier does not introduce or retain a Lean theorem.

ASSUMED-UNVERIFIED

No exhaustive historical priority claim beyond the cited Carlitz, Chow-Slattery, and OEIS records is made. No Lean formalization of this corollary is claimed.