This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340542 #31 Aug 10 2022 08:27:12
%S A340542 1,2,2,2,3,3,3,4,4,3,4,4,3,5,5,3,5,5,3,5,5,3,5,6,4,5,6,4,5,5,3,5,5,5,
%T A340542 7,5,5,7,5,3,5,5,3,7,7,3,7,8,4,5,6,4,5,7,5,5,7,5,5,5,3,7,7,5,9,7,5,7,
%U A340542 5,3,5,5,3,7,7,5,9,7,5,8,6,3,6,8,5,5,7
%N A340542 Number of Fibonacci divisors of Fibonacci(n)^2 + 1.
%C A340542 A Fibonacci divisor of a number k is a Fibonacci number that divides k.
%C A340542 It is interesting to compare this sequence with A339669.
%C A340542 We observe that a(2n) = A339669(2n) if n = 5*k + 2 or n = 5*k + 3, with k >= 0, because Lucas(2n)^2 = 5*Fibonacci(2n)^2 + 4 (see A005248: all nonnegative integer solutions of the Pell equation a(n)^2 - 5*b(n)^2 = +4 together with b(n)= A001906(n), n>=0. - _Wolfdieter Lang_, Aug 31 2004).
%C A340542 So, Lucas(2n)^2 + 1 = 5*(Fibonacci(2n)^2 + 1). Lucas(2n)^2 + 1 and Fibonacci(2n)^2 + 1 have the same Fibonacci divisors for n = 5*k + 2 or n = 5*k + 3. For the other values of n = 5*k, 5*k + 1 or 5*k + 4, 5 is a Fibonacci divisor of Lucas(2n)^2 + 1 but not of Fibonacci(2n)^2 + 1. So for these last three values of n, a(2n) = A339669(2n) - 1 (except for m = 1 and 2, 5*F(m) is never a Fibonacci number).
%H A340542 Michel Marcus, <a href="/A340542/b340542.txt">Table of n, a(n) for n = 0..200</a>
%F A340542 a(n) = A005086(A245306(n)). - _Michel Marcus_, Aug 10 2022
%e A340542 a(13) = 5 because the 5 Fibonacci divisors of Fibonacci(13)^2 + 1 = 233^2 + 1 are 1, 2, 5, 89 and 610.
%e A340542 a(16) = 5 because the 5 Fibonacci divisors of Fibonacci(16)^2 + 1 = 987^2 + 1 are 1, 2, 5, 610, and 1597.
%e A340542 Remark: the 5 Fibonacci divisors of Lucas(16)^2 + 1 = 2207^2 + 1 are 1, 2, 5, 610, and 1597, the index 16 = 2*8 with 8 of the form 5*k + 3.
%p A340542 with(combinat,fibonacci):nn:=100:F:={}:
%p A340542 for k from 0 to nn do:
%p A340542 F:=F union {fibonacci(k)}:
%p A340542 od:
%p A340542 for m from 0 to 90 do:
%p A340542 f:=fibonacci(m)^2+1:d:=numtheory[divisors](f):
%p A340542 lst:= F intersect d: n1:=nops(lst):printf(`%d, `,n1):
%p A340542 od:
%o A340542 (PARI) isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)); \\ A010056
%o A340542 a(n) = sumdiv(fibonacci(n)^2+1, d, isfib(d)); \\ _Michel Marcus_, Jan 12 2021
%Y A340542 Cf. A000032, A000045, A005248, A010056, A339461, A339669.
%Y A340542 Cf. A005086, A245306.
%K A340542 nonn
%O A340542 0,2
%A A340542 _Michel Lagneau_, Jan 12 2021