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A053031 Numbers with 1 zero in Fibonacci numbers mod m.

%I A053031 #36 Jul 30 2024 05:35:32
%S A053031 1,2,4,11,19,22,29,31,38,44,58,59,62,71,76,79,101,116,118,121,124,131,
%T A053031 139,142,151,158,179,181,191,199,202,209,211,229,236,239,242,251,262,
%U A053031 271,278,284,302,311,316,319,331,341,349,358,359,361,362,379,382,398
%N A053031 Numbers with 1 zero in Fibonacci numbers mod m.
%C A053031 Conjecture: m is on this list iff m is an odd number all of whose factors are on this list or m is 2 or 4 times such an odd number.
%C A053031 A001176(a(n)) = A128924(a(n),1) = 1. - _Reinhard Zumkeller_, Jan 16 2014
%C A053031 Also numbers n such that A001175(n) = A001177(n). - _Daniel Suteu_, Aug 08 2018
%H A053031 Reinhard Zumkeller, <a href="/A053031/b053031.txt">Table of n, a(n) for n = 1..1000</a>
%H A053031 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 A053031 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 A053031 M. Renault, <a href="http://webspace.ship.edu/msrenault/fibonacci/fib.htm">Fibonacci sequence modulo m</a>
%t A053031 With[{s = {1}~Join~Table[Count[Drop[NestWhile[Append[#, Mod[Total@ Take[#, -2], n]] &, {1, 1}, If[Length@ # < 3, True, Take[#, -2] != {1, 1}] &], -2], 0], {n, 2, 400}]}, Position[s, 1][[All, 1]] ] (* _Michael De Vlieger_, Aug 08 2018 *)
%o A053031 (Haskell)
%o A053031 a053031 n = a053031_list !! (n-1)
%o A053031 a053031_list = filter ((== 1) . a001176) [1..]
%o A053031 -- _Reinhard Zumkeller_, Jan 16 2014
%o A053031 (PARI) entryp(p)=my(k=p+[0, -1, 1, 1, -1][p%5+1], f=factor(k)); for(i=1, #f[, 1],for(j=1, f[i, 2], if((Mod([1, 1; 1, 0], p)^(k/f[i, 1]))[1, 2], break); k/=f[i, 1])); k
%o A053031 entry(n)=if(n==1, return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i, 1]>1e14, entryp(f[i, 1]^f[i, 2]), entryp(f[i, 1])*f[i, 1]^(f[i, 2]-1))); if(f[1, 1]==2&&f[1, 2]>1, v[1]=3<<max(f[1, 2]-2, 1)); lcm(v)
%o A053031 fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
%o A053031 is(n)=fibmod(entry(n)+1,n)==1 \\ _Charles R Greathouse IV_, Dec 14 2016
%Y A053031 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 A053031                              |   m=1    |   m=2   |   m=3
%Y A053031 -----------------------------+----------+---------+---------
%Y A053031 The sequence {x(n)}          | A000045  | A000129 | A006190
%Y A053031 The sequence {w(k)}          | A001176  | A214027 | A322906
%Y A053031 Primes p such that w(p) = 1  | A112860* | A309580 | A309586
%Y A053031 Primes p such that w(p) = 2  | A053027  | A309581 | A309587
%Y A053031 Primes p such that w(p) = 4  | A053028  | A261580 | A309588
%Y A053031 Numbers k such that w(k) = 1 | this seq | A309583 | A309591
%Y A053031 Numbers k such that w(k) = 2 | A053030  | A309584 | A309592
%Y A053031 Numbers k such that w(k) = 4 | A053029  | A309585 | A309593
%Y A053031 * and also A053032 U {2}
%K A053031 nonn
%O A053031 1,2
%A A053031 _Henry Bottomley_, Feb 23 2000