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

1,2

Comments

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).

Links

Table of n, a(n) for n=1..52.

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. See also Integers, (2025) Vol. 25, Art. No. A87.

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.

Example

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.

Crossrefs

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

Sequence in context: A211001 A124208 A077310 * A177051 A067130 A349496

Adjacent sequences: A381050 A381051 A381052 * A381054 A381055 A381056

Keyword

nonn

Author

Oisín Flynn-Connolly, Apr 14 2025

Extensions

More terms from Alois P. Heinz, Apr 14 2025

Status

approved