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A053029 Numbers with 4 zeros in Fibonacci numbers mod m.

%I A053029 #28 Jul 30 2024 05:35:14
%S A053029 5,10,13,17,25,26,34,37,50,53,61,65,73,74,85,89,97,106,109,113,122,
%T A053029 125,130,137,146,149,157,169,170,173,178,185,193,194,197,218,221,226,
%U A053029 233,250,257,265,269,274,277,289,293,298,305,313,314,317,325,337,338,346
%N A053029 Numbers with 4 zeros in Fibonacci numbers mod m.
%C A053029 Conjecture: m is on this list iff m is an odd number all of whose factors are on this list or m is twice such an odd number.
%C A053029 A001176(a(n)) = A128924(a(n),1) = 4. - _Reinhard Zumkeller_, Jan 17 2014
%H A053029 Reinhard Zumkeller, <a href="/A053029/b053029.txt">Table of n, a(n) for n = 1..1000</a>
%H A053029 Brennan Benfield and Michelle Manes, <a href="https://arxiv.org/abs/2202.08986">The Fibonacci Sequence is Normal Base 10</a>, arXiv:2202.08986 [math.NT], 2022.
%H A053029 Brennan Benfield and Oliver Lippard, <a href="https://arxiv.org/abs/2407.20048">Connecting Zeros in Pisano Periods to Prime Factors of K-Fibonacci Numbers</a>, arXiv:2407.20048 [math.NT], 2024.
%H A053029 M. Renault, <a href="http://webspace.ship.edu/msrenault/fibonacci/fib.htm">Fibonacci sequence modulo m</a>
%o A053029 (Haskell)
%o A053029 a053029 n = a053029_list !! (n-1)
%o A053029 a053029_list = filter ((== 4) . a001176) [1..]
%o A053029 -- _Reinhard Zumkeller_, Jan 17 2014
%Y A053029 Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
%Y A053029                              |   m=1    |   m=2   |   m=3
%Y A053029 -----------------------------+----------+---------+---------
%Y A053029 The sequence {x(n)}          | A000045  | A000129 | A006190
%Y A053029 The sequence {w(k)}          | A001176  | A214027 | A322906
%Y A053029 Primes p such that w(p) = 1  | A112860* | A309580 | A309586
%Y A053029 Primes p such that w(p) = 2  | A053027  | A309581 | A309587
%Y A053029 Primes p such that w(p) = 4  | A053028  | A261580 | A309588
%Y A053029 Numbers k such that w(k) = 1 | A053031  | A309583 | A309591
%Y A053029 Numbers k such that w(k) = 2 | A053030  | A309584 | A309592
%Y A053029 Numbers k such that w(k) = 4 | this seq | A309585 | A309593
%Y A053029 * and also A053032 U {2}
%K A053029 nonn
%O A053029 1,1
%A A053029 _Henry Bottomley_, Feb 23 2000