A237517 - OEIS

%I #43 Dec 04 2018 04:09:58

%S 1,2,3,4,5,1,7,8,9,5,11,1,13,7,15,16,17,9,19,10,21,11,23,4,25,13,27,7,

%T 29,5,31,32,33,17,35,9,37,19,39,40,41,7,43,44,45,23,47,16,49,25,17,26,

%U 53,27,55,14,19,29,59,5,61,31,63,64,65,11,67,34,23

%N Pisano period of n^2 divided by Pisano period of n.

%C For all n, a(n) | n.

%C Conjecture (Saha & Karthik): a(n) = 1 only for n = 1, 6, and 12.

%H Charles R Greathouse IV, <a href="/A237517/b237517.txt">Table of n, a(n) for n = 1..10000</a>

%H Arpan Saha and C. S. Karthik, <a href="http://arxiv.org/abs/1102.1636">A few equivalences of Wall-Sun-Sun prime conjecture</a>, arXiv:1102.1636 [math.NT], 2011.

%t pp[1] = 1; pp[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[Fibonacci[k+1], n] == 1, Return[k]]];

%t a[n_] := pp[n^2]/pp[n];

%t Array[a, 100] (* _Jean-François Alcover_, Dec 04 2018 *)

%o (PARI) fibmod(n,m)=((Mod([1,1;1,0],m))^n)[1,2]

%o entry_p(p)=my(k=1,c=Mod(1,p),o); while(c,[o,c]=[c,c+o];k++); k

%o entry(n)=if(n==1,return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i,1]>1e14,entry_p(f[i,1]^f[i,2]), entry_p(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 per(n)=if(n==1, return(1)); my(k=entry(n)); forstep(i=k,n^2,k, if(fibmod(i-1,n)==1,return(i)))

%o a(n)=per(n^2)/per(n)

%Y Cf. A001175, A001176, A237835.

%K nonn

%O 1,2

%A _Charles R Greathouse IV_, Feb 13 2014