A363742 - OEIS

%I #8 Aug 16 2023 11:03:50

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,

%T 0,1,0,0,0,1,0,1,0,0,0,0,0,2,0,0,0,0,0,1,0,1,0,0,0,3,0,0,0,1,0,1,0,0,

%U 0,1,0,4,0,0,0,0,0,1,0,2,0,0,0,4,0,0,0

%N Number of integer factorizations of n with different mean, median, and mode.

%C An integer factorization of n is a multiset of positive integers > 1 with product n.

%C If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.

%C A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

%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).

%C Position of first appearance of n is: 1, 30, 48, 60, 72, 200, 160, 96, ...

%e The a(n) factorizations for n = 30, 48, 60, 72, 96, 144:

%e   (2*3*5)  (2*3*8)    (2*5*6)    (2*4*9)    (2*6*8)    (2*8*9)

%e            (2*2*3*4)  (2*3*10)   (3*4*6)    (3*4*8)    (3*6*8)

%e                       (2*2*3*5)  (2*3*12)   (2*3*16)   (2*3*24)

%e                                  (2*2*3*6)  (2*4*12)   (2*4*18)

%e                                             (2*2*3*8)  (2*6*12)

%e                                             (2*2*4*6)  (3*4*12)

%e                                             (2*3*4*4)  (2*2*4*9)

%e                                                        (2*3*4*6)

%e                                                        (2*2*3*12)

%e                                                        (2*2*3*3*4)

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];

%t Table[Length[Select[facs[n],{Mean[#]}!={Median[#]}!=modes[#]&]],{n,100}]

%Y Just (mean) != (median): A359911, complement A359909, partitions A359894.

%Y The version for partitions is A363720, equal A363719, ranks A363730.

%Y For equal instead of unequal we have A363741.

%Y A001055 counts factorizations, strict A045778, ordered A074206.

%Y A316439 counts factorizations by length, A008284 partitions.

%Y A363265 counts factorizations with a unique mode.

%Y Cf. A089723, A237984, A240219, A326622, A339846, A339890, A359910, A362608, A363725, A363727.

%K nonn

%O 1,48

%A _Gus Wiseman_, Jun 27 2023