cp's OEIS Frontend

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.

%I A348719 #28 Feb 03 2024 10:17:09
%S A348719 1,6,12,18,20,24,30,42,48,60,120,180,240,360,420,480,504,540,630,720,
%T A348719 840,1080,1260,1440,1680,2160,2520,5040,7560,10080,15120,20160
%N 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.
%C A348719 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, ...
%e A348719 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}.
%e A348719 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}).
%t A348719 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
%o A348719 (Python)
%o A348719 from itertools import count, islice, combinations
%o A348719 from sympy import divisors
%o A348719 def A348719_gen(): # generator of terms
%o A348719     c = 0
%o A348719     yield 1
%o A348719     for n in count(2):
%o A348719         divs = tuple(divisors(n, generator=True))
%o A348719         l, b = len(divs), sum(divs)
%o A348719         if l>=4 and 2**(l-1)-l>c:
%o A348719             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)))
%o A348719             if l&1 == 0:
%o A348719                 k = l>>1
%o A348719                 m += sum(1 for p in combinations(divs,k) if 1 in p and not ((s:=sum(p))%k or (b-s)%k))
%o A348719             if m > c:
%o A348719                 yield n
%o A348719                 c = m
%o A348719 A348719_list = list(islice(A348719_gen(),10)) # _Chai Wah Wu_, Sep 24 2023
%Y A348719 Cf. A348716, A348718.
%K A348719 nonn,more
%O A348719 1,2
%A A348719 _Amiram Eldar_, Oct 31 2021
%E A348719 a(22)-a(23) from _Chai Wah Wu_, Sep 24 2023
%E A348719 a(24) from _Chai Wah Wu_, Sep 26 2023
%E A348719 a(25)-a(32) added and definition corrected by _Max Alekseyev_, Feb 01 2024