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

1,2

Comments

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

Links

Table of n, a(n) for n=1..40.

Example

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).

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]

Prog

(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

Crossrefs

Cf. A000045, A099377, A099378.

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

Sequence in context: A227963 A099678 A287687 * A290761 A192119 A050563

Adjacent sequences: A348655 A348656 A348657 * A348659 A348660 A348661

Keyword

nonn

Author

Amiram Eldar, Oct 28 2021

Status

approved