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A372048
The index of the largest Fibonacci number that divides the sum of Fibonacci numbers with indices 1 through n.
4
2, 3, 2, 2, 4, 5, 4, 4, 6, 7, 6, 6, 8, 9, 8, 8, 10, 11, 10, 10, 12, 13, 12, 12, 14, 15, 14, 14, 16, 17, 16, 16, 18, 19, 18, 18, 20, 21, 20, 20, 22, 23, 22, 22, 24, 25, 24, 24, 26, 27, 26, 26, 28, 29, 28, 28, 30, 31, 30, 30, 32, 33, 32, 32, 34, 35, 34, 34, 36, 37, 36, 36, 38, 39, 38, 38, 40, 41, 40, 40
OFFSET
1,1
COMMENTS
The sum of the first n Fibonacci numbers is sequence A000071.
When we divide the sum by the largest Fibonacci number that divides the sum, we always get a Lucas number.
For n > 3, a(n+4) = a(n)+1.
LINKS
Tanya Khovanova and the MIT PRIMES STEP senior group, Fibonacci Partial Sums Tricks, arXiv:2409.01296 [math.HO], 2024.
EXAMPLE
The sum of the first ten Fibonacci numbers is 143. The largest Fibonacci that divides this sum is 13, the seventh Fibonacci number. Thus, a(10) = 7. After the division we get 143/13 = 11, the fifth Lucas number.
MATHEMATICA
LinearRecurrence[{2, -2, 2, -1}, {2, 3, 2, 2}, 80] (* James C. McMahon, Apr 30 2024 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tanya Khovanova and the MIT PRIMES STEP senior group, Apr 17 2024
STATUS
approved