A372048 The index of the largest Fibonacci number that divides the sum of Fibonacci numbers with indices 1 through n.
2, 3, 3, 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
Examples
The sum of the first three Fibonacci numbers is 1+1+2=4. The largest Fibonacci that divides this sum is 2, the third Fibonacci number. Thus, a(3) = 2. After the division, we get 4/2 = 2, the zeroth Lucas number. 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.
Links
- Tanya Khovanova and the MIT PRIMES STEP senior group, Fibonacci Partial Sums Tricks, arXiv:2409.01296 [math.HO], 2024.
- Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Programs
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Mathematica
LinearRecurrence[{2, -2, 2, -1}, {2, 3, 3, 2, 4, 5, 4}, 80] (* James C. McMahon, Apr 30 2024, updated by Sean A. Irvine, Jul 29 2025 *)
Formula
G.f.: x*(x^6-2*x^5+2*x^4-2*x^3+x^2-x+2)/((x^2+1)*(x-1)^2). - Alois P. Heinz, Jul 25 2025
Comments