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%I #5 Jan 25 2023 09:08:55
%S 0,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1
%N Number of odd-length 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 = 120, 960, 5760, 6720:
%e 120 960 5760 6720
%e 4*5*6 2*16*30 16*18*20 4*30*56
%e 2*6*10 4*12*20 3*5*6*8*8 10*21*32
%e 8*10*12 4*4*6*6*10 12*20*28
%e 3*4*4*4*5 2*2*8*10*18 4*5*6*7*8
%e 2*2*2*4*4*5*9 2*4*7*10*12
%e 2*2*2*4*5*6*7
%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],OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,100}]
%Y The version for partitions is A359895, ranked by A359891.
%Y This is the odd-length case of A359909, partitions A240219.
%Y A001055 counts factorizations.
%Y A326622 counts factorizations with integer mean, strict A328966.
%Y Cf. A316313, A326567/A326568, A359889, A359894, A359897, A359902, A359906, A359911, A360005.
%K nonn
%O 1,8
%A _Gus Wiseman_, Jan 24 2023
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