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Numbers n such that 12*n divides F(12*n), where F(m) is the m-th Fibonacci number.
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%I #23 May 20 2019 02:49:59

%S 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,

%T 46,48,50,51,54,55,56,57,60,64,70,72,75,80,81,84,90,92,96,98,100,102,

%U 108,110,112,114,120,125,126,128,135,138,140,144,150,153,155,160,162,165

%N Numbers n such that 12*n divides F(12*n), where F(m) is the m-th Fibonacci number.

%C Odd integers n such that n divides the n-th Fibonacci number form sequence A129066, while even such n's are multiples of 12.

%C a(n) seems to be asymptotic to a constant multiple of n^phi, where phi is the golden ratio.

%H Seiichi Manyama, <a href="/A072378/b072378.txt">Table of n, a(n) for n = 1..10000</a>

%H D. Shtefan and I. Dobrovolska, <a href="https://www.fq.math.ca/Abstracts/56-3/shtefan.pdf">The sums of the consecutive Fibonacci numbers,</a>, Fibonacci Quarterly, 56 (2018), 229-236.

%e 3 belongs to the sequence because 3*12=36 divides F(36) = 14930352. For every n, 5^n belongs to the sequence, as can be proved by induction.

%t Select[Range[n], Mod[Fibonacci[12# ], 12# ]==0&]

%Y Cf. A128974, A317971.

%K nonn

%O 1,2

%A Omar Antolin Camarena (omar(AT)tlahui.posgrado.unam.mx), Jul 19 2002

%E Edited by _Max Alekseyev_, Jan 21 2010