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A053030 Numbers with 2 zeros in Fibonacci numbers mod m.

%I A053030 #31 Aug 08 2024 11:07:02
%S A053030 3,6,7,8,9,12,14,15,16,18,20,21,23,24,27,28,30,32,33,35,36,39,40,41,
%T A053030 42,43,45,46,47,48,49,51,52,54,55,56,57,60,63,64,66,67,68,69,70,72,75,
%U A053030 77,78,80,81,82,83,84,86,87,88,90,91,92,93,94,95,96,98,99,100,102,103,104
%N A053030 Numbers with 2 zeros in Fibonacci numbers mod m.
%C A053030 m is on this list iff m does not have 1 or 4 zeros in the Fibonacci sequence modulo m.
%C A053030 A001176(a(n)) = A128924(a(n),1) = 2. - _Reinhard Zumkeller_, Jan 17 2014
%H A053030 Reinhard Zumkeller, <a href="/A053030/b053030.txt">Table of n, a(n) for n = 1..10000</a>
%H A053030 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. See p. 2.
%H A053030 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 A053030 M. Renault, <a href="http://webspace.ship.edu/msrenault/fibonacci/fib.htm">Fibonacci sequence modulo m</a>
%o A053030 (Haskell)
%o A053030 a053030 n = a053030_list !! (n-1)
%o A053030 a053030_list = filter ((== 2) . a001176) [1..]
%o A053030 -- _Reinhard Zumkeller_, Jan 17 2014
%Y A053030 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 A053030                              |   m=1    |   m=2   |   m=3
%Y A053030 -----------------------------+----------+---------+---------
%Y A053030 The sequence {x(n)}          | A000045  | A000129 | A006190
%Y A053030 The sequence {w(k)}          | A001176  | A214027 | A322906
%Y A053030 Primes p such that w(p) = 1  | A112860* | A309580 | A309586
%Y A053030 Primes p such that w(p) = 2  | A053027  | A309581 | A309587
%Y A053030 Primes p such that w(p) = 4  | A053028  | A261580 | A309588
%Y A053030 Numbers k such that w(k) = 1 | A053031  | A309583 | A309591
%Y A053030 Numbers k such that w(k) = 2 | this seq | A309584 | A309592
%Y A053030 Numbers k such that w(k) = 4 | A053029  | A309585 | A309593
%Y A053030 * and also A053032 U {2}
%K A053030 nonn
%O A053030 1,1
%A A053030 _Henry Bottomley_, Feb 23 2000