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A339461 Number of Fibonacci divisors of n^2 + 1.

%I A339461 #24 Aug 10 2022 07:57:05
%S A339461 1,2,2,3,1,3,1,3,3,2,1,2,2,4,1,2,1,3,3,2,1,4,2,3,1,2,1,3,2,2,1,3,2,3,
%T A339461 3,2,1,3,2,2,1,2,2,3,2,2,1,5,2,2,1,2,2,3,1,4,1,4,2,2,2,2,2,3,1,2,1,3,
%U A339461 2,2,3,2,2,4,1,2,1,3,2,2,1,3,2,4,1,2,2
%N A339461 Number of Fibonacci divisors of n^2 + 1.
%H A339461 Michel Marcus, <a href="/A339461/b339461.txt">Table of n, a(n) for n = 0..10000</a>
%F A339461 a(A005574(n)) = 1 for n > 2.
%F A339461 a(n) = A005086(A002522(n)). - _Michel Marcus_, Dec 06 2020
%e A339461 a(13) = 4 because the divisors of 13^2 + 1 = 170 are {1, 2, 5, 10, 17, 34, 85, 170} with 4 Fibonacci divisors: 1, 2, 5 and 34.
%p A339461 with(numtheory):with(combinat,fibonacci):nn:=100:F:={}:
%p A339461 for k from 1 to nn do:
%p A339461   F:=F union {fibonacci(k)}:
%p A339461 od:
%p A339461    for n from 0 to 90 do:
%p A339461     f:=n^2+1:d:=divisors(f):
%p A339461     lst:= F intersect d: n1:=nops(lst):printf(`%d, `,n1):
%p A339461    od:
%t A339461 Array[DivisorSum[#^2 + 1, 1 &, Or @@ Map[IntegerQ@ Sqrt[#] &, 5 #^2 + 4 {-1, 1}] &] &, 105, 0] (* _Michael De Vlieger_, Dec 07 2020 *)
%o A339461 (PARI) isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || issquare(k-8);
%o A339461 a(n) = sumdiv(n^2+1, d, isfib(d)); \\ _Michel Marcus_, Dec 06 2020
%Y A339461 Cf. A000045, A002522, A005086, A005478, A005574, A128428, A338762, A338794.
%K A339461 nonn
%O A339461 0,2
%A A339461 _Michel Lagneau_, Dec 06 2020