A331977 -id:A331977 - cp's OEIS frontend

A381053 Integers k such that Fibonacci(k) is odd and divides the sum of the first Fibonacci(k) nonzero Fibonacci numbers.

1, 2, 34, 46, 68, 92, 94, 106, 166, 188, 212, 214, 226, 274, 332, 334, 346, 394, 428, 452, 454, 466, 514, 526, 548, 586, 634, 646, 668, 692, 694, 706, 754, 766, 788, 886, 908, 932, 934, 1006, 1028, 1052, 1114, 1126, 1172, 1174, 1186, 1234, 1268, 1292, 1294, 1306

Offset: 1

Oisín Flynn-Connolly, Apr 14 2025

nonn

Has infinitely many members.
Subsequence of A383021.
Contains all 2p and 4p such that p is an odd prime and p == 2,8 (mod 15).

			For k =2, Fibonacci(2) = 1, which is odd, and Fibonacci(Fibonacci(1)) = Fibonacci(1) = 1, which is divisible by 1.
For k = 34, Fibonacci(34) = 5702887 is odd, and Fibonacci(1) + Fibonacci(2) + ... + Fibonacci(5702887) = Fibonacci(5702889) - 1, which is divisible by Fibonacci(34) = 5702887.

Amirali Fatehizadeh and Daniel Yaqubi, Average of the Fibonacci numbers, J. Integer Seq. 25 (2022), no. 2, Art. 22.2.6, 10 pp.
Oisín Flynn-Connolly, On the divisibility of sums of Fibonacci numbers, arXiv:2504.09938 [math.NT], 2025.
Michal Křížek and Lawrence Somer, Period lengths modulo n and average of terms of second order linear recurrences, Integers 24 (2024), Paper No. A36, 41 pp.

Cf. A000045, A158569, A331976, A383021, A331977.

More terms from Alois P. Heinz, Apr 14 2025

A339598 Numbers k such that k | Sum_{i=1..k} A000045(i) and k+1 | Sum_{i=1..k+1} A000045(i).

Original entry on oeis.org

1, 6479, 11663, 51983, 196559

Offset: 1

Felix Fröhlich, Dec 09 2020

a(1)-a(5) are given on p. 11 in Yaqubi, Fatehizadeh, 2020. According to the authors there are no other terms up to 10^6.
Apparently an erroneous version of A331977.
Included in accordance with OEIS policy of including published but erroneous sequences to serve as pointers to the correct values.

D. Yaqubi and A. Fatehizadeh, Some results on average of Fibonacci and Lucas sequences, arXiv:2001.11839 [math.CO], 2020.

Cf. A000045, A111035.

is(n) = my(s=sum(i=1, n, fibonacci(i))); lift(Mod(s, n))==0 && lift(Mod(s+fibonacci(n+1), n+1))==0

cp's OEIS Frontend

A381053 Integers k such that Fibonacci(k) is odd and divides the sum of the first Fibonacci(k) nonzero Fibonacci numbers.

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