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

Author

Joseph L. Pe, Oct 05 2005

Keywords

• nonn

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

Examples

			2 | 4, 24 | 121392, 48 | 12586269024, ... [Corrected by _M. F. Hasler_, Feb 06 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.

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).

Programs

GAP

Filtered([1..3000], n-> ((Fibonacci(n+2)-1) mod n)=0 ); # G. C. Greubel, Feb 03 2020

Magma

[1] cat [n: n in [1..3000] | Fibonacci(n+2) mod n eq 1 ]; // G. C. Greubel, Feb 03 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 *)

PARI

is(n)=((Mod([1,1;1,0],n))^(n+2))[1,2]==1 \\ Charles R Greathouse IV, Feb 04 2013

Sage

[n for n in (1..3000) if mod(fibonacci(n+2), n)==1 ] # G. C. Greubel, Feb 03 2020

Formula

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

Extensions

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