Oisín Flynn-Connolly - 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

A383021 Self-summable Fibonacci numbers: integers k such that Fibonacci(k) divides the sum of the first Fibonacci(k) nonzero Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 12, 24, 34, 36, 46, 48, 60, 68, 72, 92, 94, 96, 106, 108, 120, 144, 166, 168, 180, 188, 192, 212, 214, 216, 226, 240, 274, 288, 300, 324, 332, 334, 336, 346, 360, 384, 394, 428, 432, 452, 454, 466, 480, 504, 514, 526, 540, 548, 552, 576, 586, 600, 612

Offset: 1

Oisín Flynn-Connolly, Apr 12 2025

nonn

Same as integers k such that Fibonacci(k) divides Fibonacci(Fibonacci(k)+2)-1.
Contains infinitely many terms.
Contains all 2p and 4p such that p is prime and p = 2,8 mod 15.
Fibonacci(k) is a subsequence of A124456.

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, A124456, A158569, A331976, A381053.

More terms from Alois P. Heinz, Apr 14 2025

A381010 Positive integers k such that 2^(k+2) - 1 is divisible by k.

Original entry on oeis.org

1, 7, 511, 713, 11023, 15553, 43873, 81079, 95263, 323593, 628153, 2275183, 6520633, 6955513, 7947583, 10817233, 12627943, 14223823, 15346303, 19852423, 27923663, 28529473, 29360527, 31019623, 39041863, 41007823, 79015273, 134217727, 143998193, 213444943, 227018383

Offset: 1

Oisín Flynn-Connolly, Apr 10 2025

nonn

7 is the only prime term.

Cf. A000225, A055685, A069927, A187787.

q:= k-> 0=(2&^(k+2)-1) mod k:
select(q, [$1..1000000])[];  # Alois P. Heinz, Apr 10 2025

isok(k) = Mod(2, k)^(k+2) == 1; \\ Michel Marcus, Apr 10 2025

def in_sequence(n):
    return pow(2, n + 2, n) == 1 % n

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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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A383021 Self-summable Fibonacci numbers: integers k such that Fibonacci(k) divides the sum of the first Fibonacci(k) nonzero Fibonacci numbers.

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A381010 Positive integers k such that 2^(k+2) - 1 is divisible by k.

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