A111058 -id:A111058 - 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

A160665 Numbers k such that the arithmetic mean of the first k Lucas numbers A000032 is an integer.

Original entry on oeis.org

1, 3, 24, 48, 72, 96, 120, 144, 192, 216, 240, 288, 336, 360, 384, 406, 432, 480, 576, 600, 648, 672, 720, 768, 864, 936, 960, 1008, 1080, 1104, 1152, 1200, 1224, 1296, 1320, 1344, 1368, 1440, 1536, 1680, 1728, 1800, 1920, 1944, 2016, 2160, 2208, 2304

Offset: 1

Ctibor O. Zizka, May 22 2009

nonn

Numbers k such that Sum_{i=0..k} A000032(i)/(k+1) is an integer. - Robert G. Wilson v, May 25 2009

Why do the terms in A141767 so closely correspond to A160665? Except for k = 1, 3, 406, 44758, 341446, 1413286, 3170242, 4861698, 7912534, ..., k == 0 (mod 24). - Robert G. Wilson v, May 25 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000032, A001610, A111058, A140826, A140971, A140972.

A000032 := proc(n) option remember ; if n <= 1 then 2-n; else procname(n-1)+procname(n-2) ; fi; end: A001610 := proc(n) add(A000032(i),i=0..n-1) ; end: for n from 1 to 3000 do if A001610(n) mod n = 0 then printf("%d,",n) ; fi; od: # R. J. Mathar, May 25 2009

lst = {}; a = 2; b = 1; s = 3; n = 3; While[n < 2447, c = a + b; s = s + c; If[Mod[c, n] == 0, AppendTo[lst, n]]; a = b; b = c; n++ ]; lst (* Robert G. Wilson v, May 25 2009 *)

{k: k | A001610(k)}. - R. J. Mathar, May 25 2009

More terms from R. J. Mathar and Robert G. Wilson v, May 25 2009

cp's OEIS Frontend

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

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