A348658 - OEIS

%I #12 Oct 28 2021 19:20:18

%S 1,3,5,6,15,21,28,140,182,496,546,672,918,1890,2016,4005,4590,24384,

%T 52780,55860,68200,84812,90090,105664,145782,186992,204600,381654,

%U 728910,907680,1655400,2302344,2862405,3828009,3926832,5959440,21059220,33550336,33839988,42325920

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

%C Terms that also Fibonacci numbers are 1, 3, 5, 21, and no more below Fibonacci(300).

%e 3 is a term since the harmonic mean of its divisors is 3/2 = Fibonacci(4)/Fibonacci(3).

%e 15 is a term since the harmonic mean of its divisors is 5/2 = Fibonacci(5)/Fibonacci(3).

%t 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 (Python)

%o from itertools import islice

%o from sympy import integer_nthroot, gcd, divisor_sigma

%o def A348658(): # generator of terms

%o     k = 1

%o     while True:

%o         a, b = divisor_sigma(k), divisor_sigma(k,0)*k

%o         c = gcd(a,b)

%o         n1, n2 = 5*(a//c)**2-4, 5*(b//c)**2-4

%o         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             yield k

%o         k += 1

%o A348658_list = list(islice(A348658(),10)) # _Chai Wah Wu_, Oct 28 2021

%Y Cf. A000045, A099377, A099378.

%Y Similar sequences: A074266, A123193, A272412, A272440, A348659.

%K nonn

%O 1,2

%A _Amiram Eldar_, Oct 28 2021