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.

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

Author

Alexander Adamchuk, Nov 02 2006, Nov 03 2006

Keywords

• nonn

Comments

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

Links

• 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.

Crossrefs

Cf. A000045, A111035, A124455.

Cf. A331976 (odd terms).

Programs

Mathematica

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

PARI

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

Sage

[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

Formula

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

Extensions

Edited by M. F. Hasler, Feb 04 2020