A111035 -id:A111035 - cp's OEIS frontend

A331977 Integers m such that m and m+1 are terms of A111035.

1, 6479, 11663, 34943, 47519, 51983, 196559, 327359, 685583, 954239, 1016063, 1346879, 2039183, 2332799, 2504447, 4665599, 5143823, 5962319, 6128639, 6723359, 7225343, 9363599, 12027023, 12446783, 14930351, 17639999, 17735759, 22924943, 24681023, 34715519, 41990399

Offset: 1

Michel Marcus, Feb 03 2020

Sequence is infinite as proved by Luca and Marques (2025). - Max Alekseyev, Aug 22 2025

Daniel Suteu, Table of n, a(n) for n = 1..128
Florian Luca and Diego Marques, On integral consecutive arithmetic means of the first Fibonacci numbers. Arch. Math. (2025).
Daniel Yaqubi and Amirali Fatehizadeh, Some results on average of Fibonacci and Lucas sequences, arXiv:2001.11839 [math.CO], 2020.

Cf. A111035 (the sum of the first k nonzero Fibonacci numbers is divisible by k).

f(n,m) = (Mod([1,1;1,0], m)^n)[1,2];
isok(n) = f(n+2,n)==1 && f(n+3, n+1)==1;
for(k=1, 10^7, if(isok(k), print1(k, ", "))); \\ Daniel Suteu, Feb 03 2020

a(23)-a(31) from Daniel Suteu, Feb 03 2020

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

A101907 Numbers k such that the arithmetic mean of the first k+1 Fibonacci numbers (beginning with F(0)) is an integer.

Original entry on oeis.org

0, 3, 5, 8, 10, 18, 23, 28, 30, 33, 40, 45, 47, 58, 60, 70, 71, 78, 88, 93, 95, 99, 100, 105, 108, 119, 128, 130, 138, 143, 148, 150, 165, 178, 180, 190, 191, 198, 200, 210, 213, 215, 219, 225, 228, 238, 239, 240, 248, 250, 268, 270, 273, 280, 287, 310, 320, 330

Offset: 1

Ctibor O. Zizka, Jul 27 2008

The sum of the first n Fibonacci numbers is F(n+2)-1, sequence A000071.

Knott discusses the factorization of these numbers. - T. D. Noe, Oct 10 2005

			k = 3 : (F(0)+F(1)+F(2)+F(3))/4 = (0+1+1+2)/4 = 1. So 3 is a term.
k = 5 : (F(0)+F(1)+F(2)+F(3)+F(4)+F(5))/6 = (0+1+1+2+3+5)/6 = 2. So 5 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., Ser. A, Vol. 81, No. 2 (2005), pp. 17-20.
H. R. Morton, Fibonacci-like sequences and greatest common divisors, The American Mathematical Monthly, Vol. 102, No. 8 (October 1995), pp. 731-734.
Morgan Ward, The prime divisors of Fibonacci numbers, Pacific J. Math., Vol. 11, No. 1 (1961), pp. 379-386.
Eric W. Weisstein's World of Mathematics, Arithmetic mean.
Eric W. Weisstein's World of Mathematics, Fibonacci Number.

Cf. A000045, A000071. See A111035 for another version.

Cf. A219612. - Altug Alkan, Dec 29 2015

Select[ Range[0, 500], Mod[Fibonacci[ # + 2] - 1, # + 1] == 0 &] (* Robert G. Wilson v *)

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

Numbers k such that (F(0)+ F(1)+ ... + F(k)) / (k+1) is an integer, where F(i) is the i-th Fibonacci number.

a(n) = A219612(n) - 1. - Altug Alkan, Dec 29 2015

Edited and extended by Robert G. Wilson v, Aug 03 2008

Definition corrected by Altug Alkan, Dec 29 2015

A331976 Odd integers m that divide the sum of the first m nonzero Fibonacci numbers.

Original entry on oeis.org

1, 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

Michel Marcus, Feb 03 2020

nonn

Not all terms are squarefree, for instance 13869, 14949, 43677, 93357, ... are not.
A subsequence of A124456, missing just the even terms A124456({2, 155, 397, 469, ...}) = {2, 758624, 7057466, 10805846, ...}. - M. F. Hasler, Feb 06 2020
There are infinitely many terms. - Oisín Flynn-Connolly, May 01 2025

Giovanni Resta, Table of n, a(n) for n = 1..10000
Oisín Flynn-Connolly, On the divisibility of sums of Fibonacci numbers, arXiv:2504.09938 [math.NT], 2025.
Daniel Yaqubi and Amirali Fatehizadeh, Some results on average of Fibonacci and Lucas sequences, arXiv:2001.11839 [math.CO], 2020.

Odd terms in A111035, A124456.

forstep(n=1,oo,2, is_A111035(n) && print1(n",")) \\ M. F. Hasler, Feb 06 2020

Equals A124456 without even terms. - M. F. Hasler, Feb 06 2020

A111058 Numbers k such that the average of the first k Lucas numbers is an integer.

Original entry on oeis.org

1, 2, 8, 12, 20, 24, 48, 60, 68, 72, 92, 96, 120, 140, 144, 188, 192, 200, 212, 216, 240, 288, 300, 332, 336, 360, 384, 428, 432, 440, 452, 480, 500, 548, 576, 600, 648, 660, 668, 672, 680, 692, 696, 720, 768, 780, 788, 812, 864, 908, 932, 960, 1008, 1028, 1052

Offset: 1

Jonathan Vos Post, Oct 07 2005

A111035 is the equivalent for Fibonacci numbers and has many elements in common with this sequence. T. D. Noe, who extended this sequence, noticed that, for some reason, 24 divides many of those k.

All terms are even except for the first term. - Harvey P. Dale, Apr 22 2024

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A000032, A000204, A027961, A111035, A160665.

Lucas[n_] := Fibonacci[n+1]+Fibonacci[n-1]; lst={}; s=0; Do[s=s+Lucas[n]; If[Mod[s, n]==0, AppendTo[lst, n]], {n, 1000}]; lst (* T. D. Noe *)
Module[{nn=1000,ln},ln=LucasL[Range[nn]];Table[If[IntegerQ[Mean[Take[ln,n]]],n,Nothing],{n,nn}]] (* Harvey P. Dale, Apr 22 2024 *)

k such that (Sum_{i=1..k} A000204(i))/k is an integer.

{ k : A027961(k) == 0 (mod k) }. - Alois P. Heinz, Apr 23 2024

A331870 Even numbers n which divide the sum of the Fibonacci numbers F(1) + ... + F(n) but are not a multiple of 24.

Original entry on oeis.org

2, 758642, 7057466, 10805846, 50860946, 59677526, 61800878, 155045678, 178551374, 217281146, 343943882, 359455694, 432175586, 609069506, 1449599486, 1721358698, 1829675354, 1884592706, 2013264194, 2116706282, 2680549946, 2971193186, 3084402122, 3252387386, 3454785386

Offset: 1

M. F. Hasler, Feb 29 2020

nonn

A111035 lists numbers n which divide the sum of the first n nonzero Fibonacci numbers. Most of these are multiples of 24. Sequence A124456 lists those which aren't. Most of these are odd (cf. A331976), this sequence lists the exceptions.
a(2) was found by Don Reble, cf. A124456.
If we consider F(n+2) = 1 + the sum of the first n nonzero Fibonacci numbers (cf. A000071), then for even n we find:
     4 divides F(n+2) for n == 4 (mod 12), 3 divides F(n+2) for n == 6 (mod 12),
  F(n+2) == 3 (mod 4) for n == 8 (mod 12), 2 divides F(n+2) for n == 10 (mod 12),
  F(n+2) == 5 (mod 6) for n == 12 (mod 24).
  These relations imply that all terms a(n) == 2 (mod 12) for all n. This also means that all terms of A111035 are either divisible by 24, or odd, or congruent to 2 (mod 12).

Giovanni Resta, Table of n, a(n) for n = 1..80

Cf. A124456, A331976, A111035, A000045 (Fibonacci numbers), A000071 (F(n)-1 = F(0)+...+F(n-2)).

M=[1,1;1,0]; forstep(n=2,oo,12,n%24&&(Mod(M,n)^(n+1))[1,1]==1&& print1(n",")) \\ Custom implementation of is_A111035(), check for updates there.

a(n) == 2 (mod 12) for all n.

Terms a(15) and beyond from Giovanni Resta, Mar 02 2020

A124455 Numbers n such that 24n divides the sum of the first 24n nonzero Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 25, 27, 28, 30, 32, 36, 40, 42, 45, 46, 48, 50, 51, 54, 55, 56, 57, 60, 64, 70, 72, 75, 80, 81, 84, 86, 90, 92, 96, 98, 100, 102, 108, 110, 112, 114, 120, 125, 126, 128, 135, 138, 140, 144, 150, 153, 155, 160, 162

Offset: 1

Alexander Adamchuk, Nov 02 2006

nonn

Numbers n such that n divides the sum of the first n nonzero Fibonacci numbers are listed in A111035 = {1, 2, 24, 48, 72, 77, 96, 120, 144, 192, 216, 240, 288, 319, 323, 336, 360, ...}. Most of these are multiples of 24. Those which are not a multiple of 24 are listed in A124456 = {1, 2, 77, 319, 323, 1517, 3021, 4757, 6479, 7221, 8159, 8229, 9797, ...}.
This sequence coincides with A072378 (12n | F(12n)) for all values up to 84. The first two different terms are 86 and 164.
Prime a(n) are {2, 3, 5, 281, ...}.

Cf. A111035, A124456, A072378 (numbers n such that 12n divides Fibonacci(12n)), A000045 (Fibonacci numbers).

Select[Range[10000], IntegerQ[ #/24]&&Mod[Fibonacci[ #+2]-1, # ]==0&] /24

Edited by M. F. Hasler, Feb 04 2020

A160757 Averages of the Fibonacci numbers which take integer values.

Original entry on oeis.org

1, 1, 5058, 262213938, 18124577012898, 187952389930860, 1409394295257361938, 116903055445824294157698, 10100618828005365858877129458, 81435914480042681825934186407384633298, 7505278652741640947693896415563573183203138, 700346071081054203480884565881868806176873272498

Offset: 1

Daniel Tisdale, May 25 2009

The numbers n such that F(1)+F(2)+...+F(n)=F(n+2)-1 is divisible by n are given in A111035. [From Max Alekseyev, Jun 04 2009]

Cf. A050248, integer average of n primes for some n, A000045.

lst = {}; Do[a = Sum[ Fibonacci@ j, {j, n}]/n; If[ IntegerQ@ a, AppendTo[lst, a]], {n, 250}]; lst

1/n*Sum {j=1..n} Fibonacci_j is an integer.

a(n) = (A000045(A111035(n)+2)-1) / A111035(n) = A000071(A111035(n)+2) / A111035(n) [From Max Alekseyev, Jun 04 2009]

Corrected and extended by Max Alekseyev and Robert G. Wilson v, Jun 04 2009

A282772 Starting from F(n), minimum number, greater than 1, of consecutive Fibonacci numbers whose average is an integer.

Original entry on oeis.org

4, 2, 3, 12, 2, 13, 3, 2, 6, 5, 2, 12, 4, 2, 3, 12, 2, 24, 3, 2, 6, 24, 2, 12, 4, 2, 3, 12, 2, 5, 3, 2, 6, 13, 2, 12, 4, 2, 3, 5, 2, 24, 3, 2, 5, 24, 2, 12, 4, 2, 3, 12, 2, 24, 3, 2, 6, 24, 2, 5, 4, 2, 3, 12, 2, 24, 3, 2, 6, 5, 2, 12, 4, 2, 3, 12, 2, 24, 3, 2, 6

Offset: 0

Paolo P. Lava, Mar 03 2017

Entries are 2, 3, 4, 5, 6, 12, 13 and 24.

Periodic with period equal to 420.

			a(0) = 4 because F(0) + F(1) + F(2) + F(3) = 0 + 1 + 1 + 2 = 4 and 4/4 = 1;
a(1) = 2 because F(1) + F(2) = 1 + 1 = 2 and 2/2 = 1;
a(2) = 3 because F(2) + F(3) + F(4) = 1 + 2 + 3 = 6 and 6/3 = 2;
a(3) = 12 because F(3) + F(4) + ... + F(13) + F(14) =  2 + 3 + ... + 233 + 377 = 984 and 984/12 = 82.

Paolo P. Lava, Table of n, a(n) for n = 0..1000

Cf. A000045, A101907, A111035, A254141.

with(combinat): P:=proc(q) local a,k,n; for k from 0 to q do a:=fibonacci(k); for n from 1 to q do a:=a+fibonacci(k+n);
if type(a/(n+1),integer) then print(n+1); break; fi; od; od; end: P(10^3);

Table[k = 1; While[! IntegerQ@ Mean@ Take[#, n ;; n + k], k++]; k + 1, {n, Length@ # - 24}] &@ Fibonacci@ Range[0, 419] (* Michael De Vlieger, Mar 06 2017 *)

a(3*k + 1) = 2;
a(12*k + 2) = a(12*k + 6) = 3;
a(12*k) = 4;
a(30*k + 9) = a(30*k + 29) = a(60*k + 44) = 5;
a(60*k + 8) = a(60*k + 20) = a(60*k + 32) = a(60*k + 56) = 6;
a(60*k + 3) = a(60*k + 11) = a(60*k + 15) = a(60*k + 23) = a(60*k + 27) = a(60*k + 35) = a(60*k + 47) = a(60*k + 51) = 12;
a(420*k + 5) = a(420*k + 33) = a(420*k + 117) = a(420*k + 173) = a(420*k + 201) = a(420*k + 257) = a(420*k + 285) = a(420*k + 341) = 13;
a(420*k + x) = 24, with x = 17, 21, 41, 45, 53, 57, 65, 77, 81, 93, 101, 105, 113, 125, 137, 141, 153, 161, 165, 177, 185, 197, 213, 221, 225, 233, 237, 245, 261, 273, 281, 293, 297, 305, 317, 321, 333, 345, 353, 365, 377, 381, 393, 401, 405, 413, 417.

A339598 Numbers k such that k | Sum_{i=1..k} A000045(i) and k+1 | Sum_{i=1..k+1} A000045(i).

Original entry on oeis.org

1, 6479, 11663, 51983, 196559

Offset: 1

Felix Fröhlich, Dec 09 2020

a(1)-a(5) are given on p. 11 in Yaqubi, Fatehizadeh, 2020. According to the authors there are no other terms up to 10^6.
Apparently an erroneous version of A331977.
Included in accordance with OEIS policy of including published but erroneous sequences to serve as pointers to the correct values.

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

Cf. A000045, A111035.

is(n) = my(s=sum(i=1, n, fibonacci(i))); lift(Mod(s, n))==0 && lift(Mod(s+fibonacci(n+1), n+1))==0

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A331977 Integers m such that m and m+1 are terms of A111035.

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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.

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A101907 Numbers k such that the arithmetic mean of the first k+1 Fibonacci numbers (beginning with F(0)) is an integer.

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A331976 Odd integers m that divide the sum of the first m nonzero Fibonacci numbers.

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A111058 Numbers k such that the average of the first k Lucas numbers is an integer.

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A331870 Even numbers n which divide the sum of the Fibonacci numbers F(1) + ... + F(n) but are not a multiple of 24.

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A124455 Numbers n such that 24n divides the sum of the first 24n nonzero Fibonacci numbers.

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A160757 Averages of the Fibonacci numbers which take integer values.

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