A363742 Number of integer factorizations of n with different mean, median, and mode.

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, 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, 0, 1, 0, 4, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0

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, ...

Links

Table of n, a(n) for n=1..87.

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

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

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

For equal instead of unequal we have A363741.

A001055 counts factorizations, strict A045778, ordered A074206.

A316439 counts factorizations by length, A008284 partitions.

A363265 counts factorizations with a unique mode.

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

Sequence in context: A088722 A336917 A122180 * A033772 A086015 A249856

Adjacent sequences: A363739 A363740 A363741 * A363743 A363744 A363745

Keyword

nonn

Author

Gus Wiseman, Jun 27 2023

Status

approved