A071774 Related to Pisano periods: integers k such that the period of Fibonacci numbers mod k equals 2*(k+1).

3, 7, 13, 17, 23, 37, 43, 53, 67, 73, 83, 97, 103, 127, 137, 157, 163, 167, 173, 193, 197, 223, 227, 257, 277, 283, 293, 313, 317, 337, 367, 373, 383, 397, 433, 443, 457, 463, 467, 487, 503, 523, 547, 577, 587, 593, 607, 613, 617, 643, 647, 653, 673, 683, 727

Offset

1,1

Comments

Terms are primes with final digit 3 or 7.

Apparently these are the primes given in A003631 without 2 and A216067. - Klaus Purath, Dec 11 2020

If k is a term, then for m=5*k the period of Fibonacci numbers mod m equals 2*(m+5). - Matthew Goers, Jan 13 2021

Links

Michael De Vlieger, Table of n, a(n) for n = 1..3969

Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.

Mathematica

Select[Prime@ Range[129], Function[n, Mod[Last@ NestWhile[{Mod[#2, n], Mod[#1 + #2, n], #3 + 1} & @@ # &, {1, 1, 1}, #[[1 ;; 2]] != {0, 1} &], n] == Mod[2 (n + 1), n] ]] (* Michael De Vlieger, Mar 31 2021, after Leo C. Stein at A001175 *)

Prog

(PARI) for(n=2, 5000, t=2*(n+1); good=1; if(fibonacci(t)%n==0, for(s=0, t, if(fibonacci(t+s)%n!=fibonacci(s)%n, good=0; break); if(s>1&&s<t-1&&fibonacci(s)%n==0, cur=s; good2=1; for(ss=0, s, if(fibonacci(ss+s)%n!=fibonacci(ss)%n, good2=0; break)); if(good2, good=0; break); ); ); if(good, print1(n, ", ")))) \\ Lambert Klasen (Lambert.Klasen(AT)gmx.net), Dec 21 2004
(PARI) forprime(p=3, 3000, if(p%5==2||p%5==3, a=1; b=0; c=1; while(a!=0||b!=1, c++; d=a; a=b; a=(a+d)%p; b=d%p); if(c==(2*(p+1)), print1(p", ")))) /* V. Raman, Nov 22 2012 */

Crossrefs

Cf. A001175, A060305, A071776.

Cf. A003631, A216067.

Sequence in context: A034913 A040148 A097957 * A019403 A045422 A191071

Adjacent sequences: A071771 A071772 A071773 * A071775 A071776 A071777

Keyword

easy,nonn

Author

Benoit Cloitre, Jun 04 2002

Extensions

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Dec 21 2004

Status

approved