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

1,2

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.

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 *)

Prog

(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<<max(f[1, 2]-2, 1)); lcm(v)
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)))
a(n)=per(n^2)/per(n)

Crossrefs

Cf. A001175, A001176, A237835.

Sequence in context: A100994 A375228 A140523 * A332883 A017666 A253247

Adjacent sequences: A237514 A237515 A237516 * A237518 A237519 A237520

Keyword

nonn

Author

Charles R Greathouse IV, Feb 13 2014

Status

approved