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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.

%I A124456 #51 Apr 18 2025 14:28:58
%S A124456 1,2,77,319,323,1517,3021,4757,6479,7221,8159,8229,9797,11663,12597,
%T A124456 13629,13869,14429,14949,16637,18407,19043,19437,23407,24947,25437,
%U A124456 30049,30621,34943,34989,35207,39203,43677,44099,47519,51983,53663,55221,65471,70221,77837,78089,79547
%N 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.
%C A124456 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]
%C A124456 A111035(2024) = 758642 is in this sequence but not in A331976. - _Don Reble_, Feb 04 2020
%C A124456 The even terms a({2, 155, 397, 469, ...}) = {2, 758642, 7057466, 10805846, ...} are now listed in A331870. - _M. F. Hasler_, Feb 06 2020
%H A124456 Giovanni Resta, <a href="/A124456/b124456.txt">Table of n, a(n) for n = 1..10000</a> (first 200 terms from M. F. Hasler)
%H A124456 Oisín Flynn-Connolly, <a href="https://arxiv.org/abs/2504.09938">On the divisibility of sums of Fibonacci numbers</a>, arXiv:2504.09938 [math.NT], 2025. See p. 3.
%F A124456 { n != 0 (mod 24) | A000071(n+2) == 0 (mod n) }. - _M. F. Hasler_, Feb 06 2020
%t A124456 Select[Range[20000], !IntegerQ[ #/24]&&Mod[Fibonacci[ #+2]-1, # ]==0&]
%o A124456 (PARI) A124456_vec(N=44, n=0)={vector(N,i, until( n++%24&& is_A111035(n),); n)} \\ _M. F. Hasler_, Feb 04 2020
%o A124456 (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
%Y A124456 Cf. A000045, A111035, A124455.
%Y A124456 Cf. A331976 (odd terms).
%K A124456 nonn
%O A124456 1,2
%A A124456 _Alexander Adamchuk_, Nov 02 2006, Nov 03 2006
%E A124456 Edited by _M. F. Hasler_, Feb 04 2020