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A348658 Numbers whose numerator and denominator of the harmonic mean of their divisors are both Fibonacci numbers.

%I A348658 #12 Oct 28 2021 19:20:18
%S A348658 1,3,5,6,15,21,28,140,182,496,546,672,918,1890,2016,4005,4590,24384,
%T A348658 52780,55860,68200,84812,90090,105664,145782,186992,204600,381654,
%U A348658 728910,907680,1655400,2302344,2862405,3828009,3926832,5959440,21059220,33550336,33839988,42325920
%N A348658 Numbers whose numerator and denominator of the harmonic mean of their divisors are both Fibonacci numbers.
%C A348658 Terms that also Fibonacci numbers are 1, 3, 5, 21, and no more below Fibonacci(300).
%e A348658 3 is a term since the harmonic mean of its divisors is 3/2 = Fibonacci(4)/Fibonacci(3).
%e A348658 15 is a term since the harmonic mean of its divisors is 5/2 = Fibonacci(5)/Fibonacci(3).
%t A348658 fibQ[n_] := Or @@ IntegerQ /@ Sqrt[{5 n^2 - 4, 5 n^2 + 4}]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := fibQ[Numerator[(hn = h[n])]] && fibQ[Denominator[hn]]; Select[Range[1000], q]
%o A348658 (Python)
%o A348658 from itertools import islice
%o A348658 from sympy import integer_nthroot, gcd, divisor_sigma
%o A348658 def A348658(): # generator of terms
%o A348658     k = 1
%o A348658     while True:
%o A348658         a, b = divisor_sigma(k), divisor_sigma(k,0)*k
%o A348658         c = gcd(a,b)
%o A348658         n1, n2 = 5*(a//c)**2-4, 5*(b//c)**2-4
%o A348658         if (integer_nthroot(n1,2)[1] or integer_nthroot(n1+8,2)[1]) and (integer_nthroot(n2,2)[1] or integer_nthroot(n2+8,2)[1]):
%o A348658             yield k
%o A348658         k += 1
%o A348658 A348658_list = list(islice(A348658(),10)) # _Chai Wah Wu_, Oct 28 2021
%Y A348658 Cf. A000045, A099377, A099378.
%Y A348658 Similar sequences: A074266, A123193, A272412, A272440, A348659.
%K A348658 nonn
%O A348658 1,2
%A A348658 _Amiram Eldar_, Oct 28 2021