OFFSET
1,4
COMMENTS
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, 2, 4, 16, 64, 5832, 4096, ...).
EXAMPLE
The factorization 6*9*9*12 = 5832 has mean 9, median 9, and modes {9}, so it is counted under a(5832).
The a(n) factorizations for selected n:
2 4 16 64 5832 4096
2*2 4*4 8*8 18*18*18 64*64
2*2*2*2 4*4*4 6*9*9*12 8*8*8*8
2*2*2*2*2*2 3*6*6*6*9 16*16*16
2*3*3*3*3*3*3*4 4*4*4*4*4*4
2*2*2*2*2*2*2*2*2*2*2*2
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 26 2023
STATUS
approved