A237835 - OEIS

%I #19 Feb 16 2025 08:33:21

%S 1,1,1,1,1,6,1,1,1,2,1,12,1,2,1,1,1,2,1,2,1,2,1,6,1,2,1,4,1,6,1,1,1,2,

%T 1,4,1,2,1,1,1,6,1,1,1,2,1,3,1,2,3,2,1,2,1,4,3,2,1,12,1,2,1,1,1,6,1,2,

%U 3,2,1,2,1,2,1,1,1,6,1,1,1,2,1,12,1,2,1

%N a(n) = n*(Pisano period of n) divided by (Pisano period of n^2).

%H Charles R Greathouse IV, <a href="/A237835/b237835.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.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PisanoPeriod.html">Pisano period</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pisano_period">Pisano period</a>.

%F a(n) = n/A237517(n).

%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_] := n pp[n]/pp[n^2];

%t Array[a, 100] (* _Jean-François Alcover_, Dec 06 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)=n*per(n)/per(n^2)

%Y Cf. A237517, A001175, A001176.

%K nonn,changed

%O 1,6

%A _Charles R Greathouse IV_, Feb 13 2014