A285058 Numbers k such that k = A104714(A285057(k)).

1, 2, 5, 7, 10, 12, 13, 17, 24, 25, 26, 29, 34, 35, 36, 37, 38, 43, 47, 48, 49, 50, 53, 55, 58, 59, 60, 65, 67, 72, 73, 74, 85, 89, 91, 96, 97, 103, 106, 108, 109, 110, 113, 118, 120, 125, 127, 130, 137, 139, 144, 145, 146, 149, 156, 157, 158, 163, 168, 169, 170, 173, 175

Offset: 1

Author

Michel Marcus, Apr 09 2017

Keywords

• nonn

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.

Crossrefs

Cf. A001177, A104714, A285057.

Programs

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 *)

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<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