A325021 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820), tau(k) is the number of divisors of k (A000005), and sigma(k) is the sum of the divisors of k (A000203).

1, 6, 28, 496, 672, 8128, 30240, 32760, 332640, 695520, 2178540, 17428320, 23569920, 33550336, 45532800, 52141320, 142990848, 164989440, 318729600, 447828480, 481572000, 500860800, 540277920, 623397600, 644271264, 714954240, 995248800, 1047254400, 1307124000

Offset: 1

Author

Jaroslav Krizek, Mar 27 2019

Keywords

• nonn

Comments

Numbers m such that m*tau(m)/sigma(m) is an integer g and simultaneously m*(m-tau(m))/sigma(m) is an integer h. Corresponding values of integers g: 1, 2, 3, 5, 8, 7, 24, 24, 44, 46, 54, 96, 80, 13, 96, ...

Corresponding values of integers h: 0, 1, 11, 243, 216, 4057, 7536, 8166, 76186, 166589, ...

Even perfect numbers from A000396 are terms.

Complement of A325022 with respect to A001599.

Intersection of A325020 and A001599.

Examples

			Harmonic number 28 is a term because 28*tau(28)/sigma(28) = 28*6/56 = 3 (integer) and simultaneously 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..255 (terms below 10^14)

Crossrefs

Cf. A000005, A000203, A001599, A007691, A049820, A325020, A325022, A325023, A325024.

Programs

Magma

[n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n) * n) / SumOfDivisors(n)) and IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n))]

Mathematica

Select[Range[10^6], And[IntegerQ@ HarmonicMean@ #2, IntegerQ[#1 (#1 - #3)/#4]] & @@ Join[{#}, {Divisors@ #}, DivisorSigma[{0, 1}, #]] &] (* Michael De Vlieger, Mar 27 2019 *)

PARI

isok(m) = my(d=numdiv(m), s=sigma(m)); !frac(m*d/s) && !frac(m*(m-d)/s); \\ Michel Marcus, Mar 27 2019

Python

from itertools import count, islice
from math import prod
from functools import reduce
from sympy import factorint
def A325021_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
        if not (n*n%s or reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s):
            yield n
A325021_list = list(islice(A325021_gen(),10)) # Chai Wah Wu, Feb 14 2023