A348719 Numbers k delivering the records in the number of partitions of divisors of k into two sets of size > 1 with an integer arithmetic mean in each set.

1, 6, 12, 18, 20, 24, 30, 42, 48, 60, 120, 180, 240, 360, 420, 480, 504, 540, 630, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 5040, 7560, 10080, 15120, 20160

Offset

1,2

Comments

The corresponding record values are 0, 1, 3, 4, 6, 16, 20, 21, 29, 198, 1542, 3448, 9055, 86081, 245107, 245145, 245336, 249208, 250595, 4844170, 31669733, ...

Links

Table of n, a(n) for n=1..32.

Example

6 is the smallest number whose set of divisors can be partitioned into two disjoint sets whose arithmetic means are both integers: {1, 3} and {2, 6}.
12 is the smallest number whose set of divisors can be partitioned into two disjoint sets whose arithmetic means are both integers in three ways: ({1, 3}, {2, 4, 6, 12}), ({2, 6}, {1, 3, 4, 12}) and ({4, 12}, {1, 2, 3, 6}).

Mathematica

q[d_] := Length[d] > 1 && IntegerQ @ Mean[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

Prog

(Python)
from itertools import count, islice, combinations
from sympy import divisors
def A348719_gen(): # generator of terms
    c = 0
    yield 1
    for n in count(2):
        divs = tuple(divisors(n, generator=True))
        l, b = len(divs), sum(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(p))%k or (b-s)%(l-k)))
            if l&1 == 0:
                k = l>>1
                m += sum(1 for p in combinations(divs, k) if 1 in p and not ((s:=sum(p))%k or (b-s)%k))
            if m > c:
                yield n
                c = m
A348719_list = list(islice(A348719_gen(), 10)) # Chai Wah Wu, Sep 24 2023

Crossrefs

Cf. A348716, A348718.

Sequence in context: A088723 A228870 A291022 * A316221 A138939 A221220

Adjacent sequences: A348716 A348717 A348718 * A348720 A348721 A348722

Keyword

nonn,more

Author

Amiram Eldar, Oct 31 2021

Extensions

a(22)-a(23) from Chai Wah Wu, Sep 24 2023

a(24) from Chai Wah Wu, Sep 26 2023

a(25)-a(32) added and definition corrected by Max Alekseyev, Feb 01 2024

Status

approved