A124455 -id:A124455 - cp's OEIS frontend

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

Joseph L. Pe, Oct 05 2005

nonn

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

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

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.

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).

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

[1] cat [n: n in [1..3000] | Fibonacci(n+2) mod n eq 1 ]; // G. C. Greubel, Feb 03 2020

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

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

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

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

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

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

A124456 Numbers k which divide the sum of the Fibonacci numbers F(1) through F(k) and such that k is not a multiple of 24.

Original entry on oeis.org

1, 2, 77, 319, 323, 1517, 3021, 4757, 6479, 7221, 8159, 8229, 9797, 11663, 12597, 13629, 13869, 14429, 14949, 16637, 18407, 19043, 19437, 23407, 24947, 25437, 30049, 30621, 34943, 34989, 35207, 39203, 43677, 44099, 47519, 51983, 53663, 55221, 65471, 70221, 77837, 78089, 79547

Offset: 1

Alexander Adamchuk, Nov 02 2006, Nov 03 2006

nonn

Numbers k which divide the sum of the first k nonzero Fibonacci numbers are listed in A111035 = {1, 2, 24, 48, 72, 77, 96, ...}. Most of these are multiples of 24. These multiples divided by 24 are listed in A124455 = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, ...}. [Edited by M. F. Hasler, Feb 04 2020]
A111035(2024) = 758642 is in this sequence but not in A331976. - Don Reble, Feb 04 2020
The even terms a({2, 155, 397, 469, ...}) = {2, 758642, 7057466, 10805846, ...} are now listed in A331870. - M. F. Hasler, Feb 06 2020

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 200 terms from M. F. Hasler)
Oisín Flynn-Connolly, On the divisibility of sums of Fibonacci numbers, arXiv:2504.09938 [math.NT], 2025. See p. 3.

Cf. A000045, A111035, A124455.

Cf. A331976 (odd terms).

Select[Range[20000], !IntegerQ[ #/24]&&Mod[Fibonacci[ #+2]-1, # ]==0&]

A124456_vec(N=44, n=0)={vector(N,i, until( n++%24&& is_A111035(n),); n)} \\ M. F. Hasler, Feb 04 2020

[n for n in (1..20000) if mod(n,24)!=0 and mod(fibonacci(n+2)-1, n)==0 ] # G. C. Greubel, Feb 16 2020

{ n != 0 (mod 24) | A000071(n+2) == 0 (mod n) }. - M. F. Hasler, Feb 06 2020

Edited by M. F. Hasler, Feb 04 2020

cp's OEIS Frontend

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

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