cp's OEIS Frontend

A214027 The number of zeros in the fundamental Pisano period of the sequence A000129 mod n.

%I A214027 #19 Aug 11 2019 01:41:23
%S A214027 1,1,2,1,4,2,1,1,2,2,2,2,4,1,2,1,2,2,2,1,2,2,1,1,4,2,2,1,4,2,1,1,2,2,
%T A214027 2,2,4,2,2,1,1,2,2,2,2,1,1,1,1,2,2,1,4,2,2,1,2,2,2,2,4,1,2,1,4,2,2,2,
%U A214027 2,2,1,1,2,2,2,2,2,2,1,1,2,1,2,2,2,2,2,1
%N A214027 The number of zeros in the fundamental Pisano period of the sequence A000129 mod n.
%C A214027 This is intimately connected with A175181 and A214028, much as A001176 is intimately connected with A001175 and A001177. In fact, A175181(n)/a(n) = A214028(n). This is the same divisibility relation that holds between A001175, A001176 and A001177.
%H A214027 T. D. Noe, <a href="/A214027/b214027.txt">Table of n, a(n) for n = 1..1000</a>
%F A214027 From _Jianing Song_, Sep 12 2018: (Start)
%F A214027 For odd primes p, a(p^e) = 4 if A214028(p) is odd; 1 if A214028(p) is even but not divisible by 4; 2 if A214028(p) is divisible by 4.
%F A214027 a(n) = 2 for n == 3 (mod 8). For primes p, a(p^e) = 1 if p == 7 (mod 8), 4 if p == 5 (mod 8). Conjecture: 1/6 of the primes congruent to 1 mod 8 satisfy a(p^e) = 1, 2/3 of them satisfy a(p^e) = 2 and 1/6 of them satisfy a(p^e) = 4.
%F A214027 (End)
%t A214027 Join[{1}, Table[s = t = Mod[{0, 1}, n]; zeros = 0; While[tmp = Mod[2*t[[2]] + t[[1]], n]; t[[1]] = t[[2]]; t[[2]] = tmp; s != t, If[tmp == 0, zeros++]]; zeros, {n, 2, 100}]] (* _T. D. Noe_, Jul 09 2012 *)
%o A214027 (PARI) A000129(m) = ([2, 1; 1, 0]^m)[2, 1]
%o A214027 a(n) = my(i=1); while(A000129(i)%n!=0, i++); znorder(Mod(A000129(i+1), n)) \\ _Jianing Song_, Aug 10 2019
%Y A214027 Cf. A175181, A214028.
%Y A214027 Similar sequences: A001176, A322906.
%K A214027 nonn
%O A214027 1,3
%A A214027 _Art DuPre_, Jul 04 2012