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

%I A285058 #25 Jul 11 2022 21:02:03
%S A285058 1,2,5,7,10,12,13,17,24,25,26,29,34,35,36,37,38,43,47,48,49,50,53,55,
%T A285058 58,59,60,65,67,72,73,74,85,89,91,96,97,103,106,108,109,110,113,118,
%U A285058 120,125,127,130,137,139,144,145,146,149,156,157,158,163,168,169,170,173,175
%N A285058 Numbers k such that k = A104714(A285057(k)).
%C A285058 The set of distinct values of A104714.
%C A285058 Leonetti & Sanna prove that this sequence is the image of A104714 for n >= 1.
%C A285058 Leonetti & Sanna prove that this sequence has density 0 and a(n) << n log n. - _Charles R Greathouse IV_, May 08 2017
%H A285058 Charles R Greathouse IV, <a href="/A285058/b285058.txt">Table of n, a(n) for n = 1..10000</a>
%H A285058 Abhishek Jha and Carlo Sanna, <a href="https://arxiv.org/abs/2207.03521">On the greatest common divisor of n and the n-th Fibonacci number, II</a>, arXiv:2207.03521 [math.NT], 2022.
%H A285058 Paolo Leonetti and Carlo Sanna, <a href="https://arxiv.org/abs/1704.00151">On the greatest common divisor of n and the nth Fibonacci number</a>, arXiv:1704.00151 [math.NT], 2017. See Lemma 2 on page 2.
%t A285058 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 *)
%o A285058 (PARI) z(n)=my(k = 1); while (fibonacci(k) % n, k++); k; \\ A001177
%o A285058 l(n) = lcm(n, z(n)); \\ A285057
%o A285058 g(n) = gcd(n, fibonacci(n)); \\ A104714
%o A285058 isok(n) = n == g(l(n));
%o A285058 (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
%o A285058 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)
%o A285058 fibmod(n,m)=((Mod([1,1;1,0],m))^n)[1,2]
%o A285058 g(n)=gcd(lift(fibmod(n,n)), n)
%o A285058 is(n)=g(lcm(z(n), n))==n \\ _Charles R Greathouse IV_, May 08 2017
%o A285058 (Python)
%o A285058 from sympy import fibonacci, gcd, lcm
%o A285058 def z(n):
%o A285058     k=1
%o A285058     while fibonacci(k)%n: k+=1
%o A285058     return k
%o A285058 def l(n): return lcm(n, z(n))
%o A285058 def g(n): return gcd(n, fibonacci(n))
%o A285058 print([n for n in range(1, 201) if n==g(l(n))]) # _Indranil Ghosh_, Apr 09 2017
%Y A285058 Cf. A001177, A104714, A285057.
%K A285058 nonn
%O A285058 1,2
%A A285058 _Michel Marcus_, Apr 09 2017