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A349687 Numbers whose numerator and denominator of their abundancy index are both Fibonacci numbers.

%I A349687 #17 Nov 30 2021 04:54:55
%S A349687 1,2,6,15,24,26,28,84,90,96,120,270,330,496,672,1335,1488,1540,1638,
%T A349687 8128,24384,27280,44109,68200,131040,447040,523776,18506880,22256640,
%U A349687 33550336,36197280,38257095,65688320,91963648,95472000,100651008,102136320,176432256,197308800
%N A349687 Numbers whose numerator and denominator of their abundancy index are both Fibonacci numbers.
%C A349687 This sequence includes all the perfect numbers (A000396), 3-perfect numbers (A005820) and 5-perfect numbers (A046060).
%C A349687 The deficient terms, 1, 2, 15, 26, 1335, 44109, 38257095, ..., have an abundancy index which is a ratio of two consecutive Fibonacci numbers, 1/1, 3/2, 8/5, 21/13, 144/89, 610/377, 46368/28657, ..., which approaches the golden ratio phi = 1.618... (A001622) as the numerators and denominators get larger.
%H A349687 Michel Marcus, <a href="/A349687/a349687.txt">85 terms</a> (some terms might be missing in this list).
%e A349687 2 is a term since sigma(2)/2 = 3/2 = Fibonacci(4)/Fibonacci(3).
%e A349687 15 is a term since sigma(15)/15 = 8/5 = Fibonacci(6)/Fibonacci(5).
%t A349687 fibQ[n_] := Or @@ IntegerQ /@ Sqrt[{5 n^2 - 4, 5 n^2 + 4}]; ai[n_] := DivisorSigma[1, n]/n; q[n_] := fibQ[Numerator[(ain = ai[n])]] && fibQ[Denominator[ain]]; Select[Range[10^6], q]
%o A349687 (PARI) isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
%o A349687 isok(n) = my(q=sigma(n)/n); isfib(numerator(q)) && isfib(denominator(q)); \\ _Michel Marcus_, Nov 25 2021
%Y A349687 Cf. A000045, A010056, A000203, A001622, A017665, A017666.
%Y A349687 Subsequences: A000396, A005820, A046060.
%Y A349687 Similar sequences: A069070, A216780, A247086, A348658.
%K A349687 nonn
%O A349687 1,2
%A A349687 _Amiram Eldar_, Nov 25 2021