bibkey: lagneau2020a339669
authors: Michel Lagneau
year: 2020
title: "OEIS A339669, Number of Fibonacci divisors of Lucas(n)^2 + 1"
doi: null
url: https://oeis.org/A339669
claim: "A339669 %N: Number of Fibonacci divisors of Lucas(n)^2 + 1. A339669 %C: Particular attention must be paid to the regularity properties of the number of divisors of Lucas(n)^2 + 1 observed for n < 156, when a(n) = 1 or 2. From this observation, we propose two conjectures verified for n < 156. A339669 %C: Conjecture 1: a(6*n+3) = 1. A339669 %C: Conjecture 2: a(6*n+1) = a(6*n+5) = 2. A339669 %A: _Michel Lagneau_, Dec 12 2020"
strata_touched:
  - D5/S1/Scale/Lucas
license: citation-only
triage: anchor

OEIS A339669

The NAME of A339669 states:

Number of Fibonacci divisors of Lucas(n)^2 + 1.

The COMMENTS observation is printed as:

Particular attention must be paid to the regularity properties of the number of divisors of Lucas(n)^2 + 1 observed for n < 156, when a(n) = 1 or 2. From this observation, we propose two conjectures verified for n < 156.

The two numbered conjectures are printed as:

Conjecture 1: a(6*n+3) = 1.

Conjecture 2: a(6n+1) = a(6n+5) = 2.

The AUTHOR line is:

Michel Lagneau, Dec 12 2020

The EXAMPLE line fixes the counting convention: it lists five Fibonacci divisors of Lucas(8)^2 + 1 = 2210, namely 1, 2, 5, 13 and 34. The value 1 is counted once even though both F_1 and F_2 equal 1, so the sequence counts distinct divisor values. The same line shows that the two conjectures are restricted to odd indices for a reason: at the even index 8 the count is 5.

Verified locator

• URL: https://oeis.org/A339669
• OFFSET (verbatim): 0,1
• NAME (verbatim): Number of Fibonacci divisors of Lucas(n)^2 + 1.
• COMMENTS observation line (verbatim): Particular attention must be paid to the regularity properties of the number of divisors of Lucas(n)^2 + 1 observed for n < 156, when a(n) = 1 or 2. From this observation, we propose two conjectures verified for n < 156.
• COMMENTS Conjecture 1 line (verbatim): Conjecture 1: a(6*n+3) = 1.
• COMMENTS Conjecture 2 line (verbatim): Conjecture 2: a(6n+1) = a(6n+5) = 2.
• EXAMPLE line (verbatim): a(8) = 5 because the divisors of Lucas(8)^2 + 1 = 47^2 + 1 = 2210 are {1, 2, 5, 10, 13, 17, 26, 34, 65, 85, 130, 170, 221, 442, 1105, 2210} with 5 Fibonacci divisors: 1, 2, 5, 13 and 34.
• AUTHOR (verbatim): Michel Lagneau, Dec 12 2020
• Read on 2026-09-14 through oeis.org/search?q=id:A339669&fmt=text.