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A340542 Number of Fibonacci divisors of Fibonacci(n)^2 + 1. 1

%I #31 Aug 10 2022 08:27:12

%S 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 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 5,3,5,5,3,7,7,5,9,7,5,8,6,3,6,8,5,5,7

%N Number of Fibonacci divisors of Fibonacci(n)^2 + 1.

%C A Fibonacci divisor of a number k is a Fibonacci number that divides k.

%C It is interesting to compare this sequence with A339669.

%C 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 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 Michel Marcus, <a href="/A340542/b340542.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = A005086(A245306(n)). - _Michel Marcus_, Aug 10 2022

%e a(13) = 5 because the 5 Fibonacci divisors of Fibonacci(13)^2 + 1 = 233^2 + 1 are 1, 2, 5, 89 and 610.

%e a(16) = 5 because the 5 Fibonacci divisors of Fibonacci(16)^2 + 1 = 987^2 + 1 are 1, 2, 5, 610, and 1597.

%e 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 with(combinat,fibonacci):nn:=100:F:={}:

%p for k from 0 to nn do:

%p F:=F union {fibonacci(k)}:

%p od:

%p for m from 0 to 90 do:

%p f:=fibonacci(m)^2+1:d:=numtheory[divisors](f):

%p lst:= F intersect d: n1:=nops(lst):printf(`%d, `,n1):

%p od:

%o (PARI) isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)); \\ A010056

%o a(n) = sumdiv(fibonacci(n)^2+1, d, isfib(d)); \\ _Michel Marcus_, Jan 12 2021

%Y Cf. A000032, A000045, A005248, A010056, A339461, A339669.

%Y Cf. A005086, A245306.

%K nonn

%O 0,2

%A _Michel Lagneau_, Jan 12 2021

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Last modified March 29 06:15 EDT 2024. Contains 371265 sequences. (Running on oeis4.)