OFFSET
1,48
COMMENTS
An integer factorization of n is a multiset of positive integers > 1 with product n.
If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
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}.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
Position of first appearance of n is: 1, 30, 48, 60, 72, 200, 160, 96, ...
EXAMPLE
The a(n) factorizations for n = 30, 48, 60, 72, 96, 144:
(2*3*5) (2*3*8) (2*5*6) (2*4*9) (2*6*8) (2*8*9)
(2*2*3*4) (2*3*10) (3*4*6) (3*4*8) (3*6*8)
(2*2*3*5) (2*3*12) (2*3*16) (2*3*24)
(2*2*3*6) (2*4*12) (2*4*18)
(2*2*3*8) (2*6*12)
(2*2*4*6) (3*4*12)
(2*3*4*4) (2*2*4*9)
(2*3*4*6)
(2*2*3*12)
(2*2*3*3*4)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Table[Length[Select[facs[n], {Mean[#]}!={Median[#]}!=modes[#]&]], {n, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 27 2023
STATUS
approved