%I #8 Jan 25 2023 09:09:00
%S 0,1,1,2,1,2,1,3,2,2,1,3,1,2,2,4,1,3,1,3,2,2,1,5,2,2,3,3,1,4,1,4,2,2,
%T 2,6,1,2,2,4,1,4,1,3,3,2,1,6,2,3,2,3,1,4,2,4,2,2,1,7,1,2,3,7,2,4,1,3,
%U 2,4,1,7,1,2,3,3,2,4,1,6,4,2,1,6,2,2,2
%N Number of integer factorizations of n into factors > 1 with the same mean as median.
%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%e The a(n) factorizations for n = 24, 36, 60, 120, 144, 360:
%e 24 36 60 120 144 360
%e 3*8 4*9 2*30 2*60 2*72 4*90
%e 4*6 6*6 3*20 3*40 3*48 5*72
%e 2*12 2*18 4*15 4*30 4*36 6*60
%e 2*3*4 3*12 5*12 5*24 6*24 8*45
%e 2*2*3*3 6*10 6*20 8*18 9*40
%e 3*4*5 8*15 9*16 10*36
%e 10*12 12*12 12*30
%e 4*5*6 2*2*6*6 15*24
%e 2*6*10 3*3*4*4 18*20
%e 2*3*4*5 2*180
%e 3*120
%e 2*10*18
%e 3*4*5*6
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Table[Length[Select[facs[n],Mean[#]==Median[#]&]],{n,100}]
%Y The version for partitions is A240219, complement A359894.
%Y These multisets are ranked by A359889.
%Y The version for strict partitions is A359897.
%Y The odd-length case is A359910.
%Y The complement is counted by A359911.
%Y A001055 counts factorizations.
%Y A058398 counts partitions by mean, see also A008284, A327482.
%Y A326622 counts factorizations with integer mean, strict A328966.
%Y A359893 and A359901 count partitions by median, odd-length A359902.
%Y Cf. A316313, A326567/A326568, A359906, A360005.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jan 24 2023