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Number of integer factorizations of n into factors > 1 with the same mean as median.
9

%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