A344309 a(n) is the number of eigensequences (including eigenvalues, where they exist) of the Fibonacci recurrence in Zp where p is prime(n).

0, 0, 1, 0, 2, 1, 1, 2, 0, 3, 2, 1, 3, 0, 2, 1, 2, 5, 0, 2, 1, 2, 0, 9, 1, 3, 0, 2, 5, 5, 0, 2, 1, 4, 5, 4, 1, 0, 0, 1, 2, 3, 2, 1, 1, 10, 6, 0, 0, 3, 17, 2, 3, 2, 1, 2, 5, 2, 1, 11, 0, 1, 6, 2, 1, 1, 4, 1, 2, 3, 5, 2, 0, 1, 2, 0, 5, 1, 5, 3, 2, 21, 2, 1, 2, 0

Offset

1,5

Comments

It appears that the indices m where a(m) = 0 give A270532.

Links

Table of n, a(n) for n=1..86.

M. Aoki, Y. Sakai, On Equivalence Classes of Generalized Fibonacci Sequences, JIS vol 19 (2016) # 16.2.6

H. Sedaghat, Zero-Avoiding Solutions of the Fibonacci Recurrence Modulo A Prime, Fibonacci Quart. 52 (2014), no. 1, 39-45. See p. 44.

Formula

a(n) = E(p)/(p-1) where p is prime(n) and E(p) = 4 if p=5; (p-1)^2/ep(n) + p - 1 if p==1 or 4 (mod 5); (p^2-1)/ep(n) + 1 - p if p==2 or 3 (mod 5), where ep(n) = A001602(n).

Maple

A344309 := proc(i)
    local p, F ;
    p := ithprime(i) ;
    for n from 1 do
        F := combinat[fibonacci](n) ;
        if modp(F, p) =0 then
            if modp(p, 5) in {2, 3} then
                return (p+1)/n-1 ;
            elif modp(p, 5) in {1, 4} then
                return (p-1)/n+1 ;
            else
                return 1 ;
            end if;
        end if;
    end do:
end proc;
seq(A344309(n), n=1..55) ; # R. J. Mathar, Feb 27 2023

Prog

(PARI) ep(n) = if(n==3, 5, my(p=prime(n)); fordiv(p^2-1, d, if(fibonacci(d)%p==0, return(d)))); \\ A001602
a(n) = {my(p=prime(n), mp = p % 5); my(x=if ((mp==0), 4, if ((mp==1) || (mp==4), (p-1)^2/ep(n) + p - 1, (p^2-1)/ep(n) + 1 - p))); x/(p-1); }

Crossrefs

Cf. A001602, A270532.

Sequence in context: A374191 A088226 A376542 * A358338 A244658 A117586

Adjacent sequences: A344306 A344307 A344308 * A344310 A344311 A344312

Keyword

nonn

Author

Michel Marcus, May 14 2021

Status

approved