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

1, 3, 5, 6, 15, 21, 28, 140, 182, 496, 546, 672, 918, 1890, 2016, 4005, 4590, 24384, 52780, 55860, 68200, 84812, 90090, 105664, 145782, 186992, 204600, 381654, 728910, 907680, 1655400, 2302344, 2862405, 3828009, 3926832, 5959440, 21059220, 33550336, 33839988, 42325920

Offset: 1

Author

Amiram Eldar, Oct 28 2021

Keywords

• nonn

Comments

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

Examples

			3 is a term since the harmonic mean of its divisors is 3/2 = Fibonacci(4)/Fibonacci(3).
15 is a term since the harmonic mean of its divisors is 5/2 = Fibonacci(5)/Fibonacci(3).
		

Crossrefs

Cf. A000045, A099377, A099378.

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

Programs

Mathematica

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]

Python

from itertools import islice
from sympy import integer_nthroot, gcd, divisor_sigma
def A348658(): # generator of terms
    k = 1
    while True:
        a, b = divisor_sigma(k), divisor_sigma(k,0)*k
        c = gcd(a,b)
        n1, n2 = 5*(a//c)**2-4, 5*(b//c)**2-4
        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]):
            yield k
        k += 1
A348658_list = list(islice(A348658(),10)) # Chai Wah Wu, Oct 28 2021