A363265 Number of integer factorizations of n with a unique mode.

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 6, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Jun 27 2023

Keywords

• nonn

Comments

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

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}.

Conjecture: 9 is missing from this sequence.

Examples

			The a(n) factorizations for n = 2, 4, 16, 24, 48, 72:
  (2)  (4)    (16)       (24)       (48)         (72)
       (2*2)  (4*4)      (2*2*6)    (3*4*4)      (2*6*6)
              (2*2*4)    (2*2*2*3)  (2*2*12)     (3*3*8)
              (2*2*2*2)             (2*2*2*6)    (2*2*18)
                                    (2*2*3*4)    (2*2*2*9)
                                    (2*2*2*2*3)  (2*2*3*6)
                                                 (2*3*3*4)
                                                 (2*2*2*3*3)
		

Crossrefs

The complement for partitions is A362607, ranks A362605.

The version for partitions is A362608, ranks A356862.

A001055 counts factorizations, strict A045778, ordered A074206.

A089723 counts constant factorizations.

A316439 counts factorizations by length, A008284 partitions.

A339846 counts even-length factorizations, A339890 odd-length.

Cf. A240219, A326622, A333487, A335434, A347438, A362610, A362611, A362612, A362614, A363723.

Programs

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],Length[modes[#]]==1&]],{n,100}]