A237835 a(n) = n*(Pisano period of n) divided by (Pisano period of n^2).

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, 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, 3, 2, 1, 2, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 12, 1, 2, 1

Offset: 1

Author

Charles R Greathouse IV, Feb 13 2014

Keywords

• nonn

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Arpan Saha and C. S. Karthik, A few equivalences of Wall-Sun-Sun prime conjecture, arXiv:1102.1636 [math.NT], 2011.
• Eric Weisstein's World of Mathematics, Pisano period.
• Wikipedia, Pisano period.

Crossrefs

Cf. A237517, A001175, A001176.

Programs

Mathematica

pp[1] = 1; pp[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[Fibonacci[k+1], n] == 1, Return[k]]];
a[n_] := n pp[n]/pp[n^2];
Array[a, 100] (* Jean-François Alcover, Dec 06 2018 *)

PARI

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
entry_p(p)=my(k=1, c=Mod(1, p), o); while(c, [o, c]=[c, c+o]; k++); k
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<
				

Formula

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