OFFSET
1,2
COMMENTS
The set of distinct values of A104714.
Leonetti & Sanna prove that this sequence is the image of A104714 for n >= 1.
Leonetti & Sanna prove that this sequence has density 0 and a(n) << n log n. - Charles R Greathouse IV, May 08 2017
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Abhishek Jha and Carlo Sanna, On the greatest common divisor of n and the n-th Fibonacci number, II, arXiv:2207.03521 [math.NT], 2022.
Paolo Leonetti and Carlo Sanna, On the greatest common divisor of n and the nth Fibonacci number, arXiv:1704.00151 [math.NT], 2017. See Lemma 2 on page 2.
MATHEMATICA
z[n_]:=Block[{k=1}, While[Mod[Fibonacci[k], n]!=0, k ++]; k]; l[n_]:=LCM[n, z[n]]; g[n_]:= GCD[n, Fibonacci[n]]; Select[Range[200], #==g[l[#]] &] (* Indranil Ghosh, Apr 09 2017 *)
PROG
(PARI) z(n)=my(k = 1); while (fibonacci(k) % n, k++); k; \\ A001177
l(n) = lcm(n, z(n)); \\ A285057
g(n) = gcd(n, fibonacci(n)); \\ A104714
isok(n) = n == g(l(n));
(PARI) zp(p)=my(k=p+[0, -1, 1, 1, -1][p%5+1], f=factor(k)); for(i=1, #f[, 1], for(j=1, f[i, 2], if((Mod([1, 1; 1, 0], p)^(k/f[i, 1]))[1, 2], break); k/=f[i, 1])); k
z(n)=if(n==1, return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i, 1]>1e14, zp(f[i, 1]^f[i, 2]), zp(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)
fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
g(n)=gcd(lift(fibmod(n, n)), n)
is(n)=g(lcm(z(n), n))==n \\ Charles R Greathouse IV, May 08 2017
(Python)
from sympy import fibonacci, gcd, lcm
def z(n):
k=1
while fibonacci(k)%n: k+=1
return k
def l(n): return lcm(n, z(n))
def g(n): return gcd(n, fibonacci(n))
print([n for n in range(1, 201) if n==g(l(n))]) # Indranil Ghosh, Apr 09 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Apr 09 2017
STATUS
approved