

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.


0



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
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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



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**(l1)l>c:
m = sum(1 for k in range(2, (l1>>1)+1) for p in combinations(divs, k) if not ((s:=sum(p))%k or (bs)%(lk)))
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 (bs)%k))
if m > c:
yield n
c = m


CROSSREFS



KEYWORD

nonn,more


AUTHOR



EXTENSIONS

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


STATUS

approved



