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A237517 Pisano period of n^2 divided by Pisano period of n. 5
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A241479 A100994 A140523 * A332883 A017666 A253247

Adjacent sequences:  A237514 A237515 A237516 * A237518 A237519 A237520

KEYWORD

nonn

AUTHOR

Charles R Greathouse IV, Feb 13 2014

STATUS

approved

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Last modified November 27 04:58 EST 2021. Contains 349346 sequences. (Running on oeis4.)