A111035 Numbers n that divide the sum of the first n nonzero Fibonacci numbers.

1, 2, 24, 48, 72, 77, 96, 120, 144, 192, 216, 240, 288, 319, 323, 336, 360, 384, 432, 480, 576, 600, 648, 672, 720, 768, 864, 960, 1008, 1080, 1104, 1152, 1200, 1224, 1296, 1320, 1344, 1368, 1440, 1517, 1536, 1680, 1728, 1800, 1920, 1944, 2016, 2064, 2160

Offset

1,2

Comments

The sum of the first n nonzero Fibonacci numbers is F(n+2)-1, sequence A000071. Knott discusses the factorization of these numbers. Most of the terms are divisible by 24. - T. D. Noe, Oct 10 2005, edited by M. F. Hasler, Mar 01 2020

All terms are either multiples of 24 (cf. A124455) or odd (cf. A331976) or congruent to 2 (mod 12), cf. A331870 where this statement is proved. - M. F. Hasler, Mar 01 2020

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Ron Knott, The Mathematical Magic of the Fibonacci Numbers

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

Formula

{n: n| A000071(n+2)}. - R. J. Mathar, Feb 05 2020

Example

2 | 4, 24 | 121392, 48 | 12586269024, ... [Corrected by M. F. Hasler, Feb 06 2020]

Maple

select(n-> irem(combinat[fibonacci](n+2)-1, n)=0, [$1..3000])[]; # G. C. Greubel, Feb 03 2020

Mathematica

Select[Range[3000], Mod[Fibonacci[ #+2]-1, # ]==0&] (*  T. D. Noe, Oct 06 2005 *)

Prog

(PARI) is(n)=((Mod([1, 1; 1, 0], n))^(n+2))[1, 2]==1 \\ Charles R Greathouse IV, Feb 04 2013
(Magma) [1] cat [n: n in [1..3000] | Fibonacci(n+2) mod n eq 1 ]; // G. C. Greubel, Feb 03 2020
(Sage) [n for n in (1..3000) if mod(fibonacci(n+2), n)==1 ] # G. C. Greubel, Feb 03 2020
(GAP) Filtered([1..3000], n-> ((Fibonacci(n+2)-1) mod n)=0 ); # G. C. Greubel, Feb 03 2020

Crossrefs

See A101907 for another version.

Cf. A111058 (the analog for Lucas numbers).

Cf. A124455 (k for a(n) = 24k), A124456 (other a(n)), A331976 (odd a(n)), A331870 (even a(n) != 24k).

Sequence in context: A181283 A368360 A073215 * A349188 A249277 A002552

Adjacent sequences: A111032 A111033 A111034 * A111036 A111037 A111038

Keyword

nonn

Author

Joseph L. Pe, Oct 05 2005

Extensions

More terms from Rick L. Shepherd and T. D. Noe, Oct 06 2005

Status

approved