A237517 Pisano period of n^2 divided by Pisano period of n.

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, 29, 5, 31, 32, 33, 17, 35, 9, 37, 19, 39, 40, 41, 7, 43, 44, 45, 23, 47, 16, 49, 25, 17, 26, 53, 27, 55, 14, 19, 29, 59, 5, 61, 31, 63, 64, 65, 11, 67, 34, 23

Offset: 1

Author

Charles R Greathouse IV, Feb 13 2014

Keywords

• nonn

Comments

For all n, a(n) | n.

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

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.

Crossrefs

Cf. A001175, A001176, A237835.

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_] := pp[n^2]/pp[n];
Array[a, 100] (* Jean-François Alcover, Dec 04 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<