A348716 Numbers whose divisors can be partitioned into two disjoint sets without singletons whose harmonic means are both integers in a record number of ways.

1, 12, 24, 60, 84, 120, 240, 360, 420, 672, 840, 1260, 1680, 2160, 2520, 3360, 6720, 7560, 10080, 15120, 21840, 27720, 30240, 50400, 60480, 65520, 83160, 98280, 110880, 131040

Offset: 1

Author

Amiram Eldar, Oct 31 2021

Keywords

• nonn
• more

Comments

The corresponding record values are 0, 1, 3, 10, 26, 198, 1093, 7035, 12391, 17625, ...

From David A. Corneth, Sep 25 2023: (Start)

By multiplying the denominator of the harmonic mean of a(n) by a(n) we get a partition over the divisors of m = a(n) where two expressions must be an integer at the same time. Let v1 and v2 be the of divisors of a(n) into two disjoint sets, |v1| and |v2| their length and h1 and h2 the sum of their respective reciprocals.

Let tau(m) be the number of divisors of m (cf. A000005) and sigma(m) be the sum of divisors of m (cf. A000203). Then by definition |v1|/h1 and |v2|/h2 are integers.

Multiply the numerators and denominators by m and we get (|v1|*m)/(h1*m) and (|v2|*m)/(h2*m) both integers. Furthermore h1*m and h2*m are integers and |v1| + |v2| = tau(m) and h1*m + h2*m = sigma(m).

Substituting |v2| = tau(m) - |v1| and h2*m = sigma(m) - h1*m in (|v2|*m)/(h2*m) we get (|v1|*m)/(h1*m) and ((tau(m) - |v1|) * m) / (sigma(m) - h1*m) are positive integers where tau(m) and sigma(m) are fixed and 2 <= |v1| <= tau(m)-2 and 3 <= h1 <= sigma(m) - 3 as the smallest possible sum of two divisors is 1 + 2 = 3

The resulting pairs (|v1|, h1*m) yield separate partition problems where one should be careful with the case (if it occurs) (2*|v1|, 2*h1*m) = (tau(n), sigma(n)). (End)

Examples

			12 is the smallest number whose set of divisors can be partitioned into two disjoint sets whose harmonic means are both integers: {1, 2, 3, 6} and {4, 12}.
24 is the smallest number whose set of divisors can be partitioned into two disjoint sets whose harmonic means are both integers in three ways: ({3, 6}, {1, 2, 4, 8, 12, 24}), ({1, 3, 6}, {2, 4, 8, 12, 24}) and ({1, 2, 3, 6}, {4, 8, 12, 24}).
From _David A. Corneth_, Sep 25 2023: (Start)
For n = 24 we have tau(n) = tau(24) = 8 and sigma(n) = sigma(24) = 60.
Therefore we look for (|v1|*24) / (h1*24) and ((8-|v1|)*24) / (60 - h1*24) both integers. It turns out that the pairs (|v1|, 24*h1) in {(2, 12), (2, 24), (2, 48), (3, 36), (4, 12)} (omitting conjugates) give integers.
The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. (2, 12) gives the sum 4 + 8; a sum of two distinct divisors of 24 summing to 12. (2, 24) gives no solution, likewise (2, 48) does not. From (3, 36) we get 4 + 8 + 24 and from (4, 12) we get 1 + 2 + 3 + 6.
And indeed these correspond to 2/(8/24 + 4/24) = 2/(1/3 + 1/6), 3/(24/24 + 8/24 + 4/24) = 3/(1 + 1/3 + 1/6) and 4/(24/24 + 12/24 + 8/24 + 4/24) = 4/(1 + 1/2 + 1/3 + 1/6). (End)
		

Crossrefs

Cf. A348715.

Programs

Mathematica

q[d_] := Length[d] > 1 && IntegerQ@HarmonicMean[d]; c[n_] := Count[Subsets[(d = Divisors[n])], _?(q[#] && q[Complement[d, #]] &)]/2; cm = -1; s = {}; Do[If[(c1 = c[n]) > cm, cm = c1; AppendTo[s, n]], {n, 1, 240}]; s

Python

from fractions import Fraction
from itertools import count, islice, combinations
from sympy import divisors
def A348716_gen(): # generator of terms
    c = 0
    yield 1
    for n in count(2):
        divs = tuple(divisors(n, generator=True))
        l, b = len(divs), sum(Fraction(1,d) for d in divs)
        if l>=4 and 2**(l-1)-l>c:
            m = sum(1 for k in range(2,(l-1>>1)+1) for p in combinations(divs,k) if not ((s:=sum(Fraction(1,d) for d in p)).denominator*k%(s.numerator) or (r:=b-s).denominator*(l-k)%(r.numerator)))
            if l&1 == 0:
                k = l>>1
                m += sum(1 for p in combinations(divs,k) if 1 in p and not ((s:=sum(Fraction(1,d) for d in p)).denominator*k%(s.numerator) or (r:=b-s).denominator*k%(r.numerator)))
            if m > c:
                yield n
                c = m
A348716_list = list(islice(A348716_gen(),5)) # Chai Wah Wu, Sep 24 2023

Extensions

a(11) from Chai Wah Wu, Sep 25 2023

a(12)-a(27) from David A. Corneth, Sep 25 2023

a(28)-a(30) from Max Alekseyev, Feb 03 2024