%I #10 Aug 16 2023 11:04:11

%S 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,

%T 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,

%U 1,1,1,8,1,1,2,2,1,1,1,6,4,1,1,3,1,1,1

%N Number of integer factorizations of n with a unique mode.

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

%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 Conjecture: 9 is missing from this sequence.

%e The a(n) factorizations for n = 2, 4, 16, 24, 48, 72:

%e (2) (4) (16) (24) (48) (72)

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

%e (2*2*4) (2*2*2*3) (2*2*12) (3*3*8)

%e (2*2*2*2) (2*2*2*6) (2*2*18)

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

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

%e (2*3*3*4)

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

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

%Y The complement for partitions is A362607, ranks A362605.

%Y The version for partitions is A362608, ranks A356862.

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

%Y A089723 counts constant factorizations.

%Y A316439 counts factorizations by length, A008284 partitions.

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

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

%K nonn

%O 1,4

%A _Gus Wiseman_, Jun 27 2023