cp's OEIS Frontend

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

%I A363265 #10 Aug 16 2023 11:04:11
%S A363265 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 A363265 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 A363265 1,1,1,8,1,1,2,2,1,1,1,6,4,1,1,3,1,1,1
%N A363265 Number of integer factorizations of n with a unique mode.
%C A363265 An integer factorization of n is a multiset of positive integers > 1 with product n.
%C A363265 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 A363265 Conjecture: 9 is missing from this sequence.
%e A363265 The a(n) factorizations for n = 2, 4, 16, 24, 48, 72:
%e A363265   (2)  (4)    (16)       (24)       (48)         (72)
%e A363265        (2*2)  (4*4)      (2*2*6)    (3*4*4)      (2*6*6)
%e A363265               (2*2*4)    (2*2*2*3)  (2*2*12)     (3*3*8)
%e A363265               (2*2*2*2)             (2*2*2*6)    (2*2*18)
%e A363265                                     (2*2*3*4)    (2*2*2*9)
%e A363265                                     (2*2*2*2*3)  (2*2*3*6)
%e A363265                                                  (2*3*3*4)
%e A363265                                                  (2*2*2*3*3)
%t A363265 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A363265 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363265 Table[Length[Select[facs[n],Length[modes[#]]==1&]],{n,100}]
%Y A363265 The complement for partitions is A362607, ranks A362605.
%Y A363265 The version for partitions is A362608, ranks A356862.
%Y A363265 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A363265 A089723 counts constant factorizations.
%Y A363265 A316439 counts factorizations by length, A008284 partitions.
%Y A363265 A339846 counts even-length factorizations, A339890 odd-length.
%Y A363265 Cf. A240219, A326622, A333487, A335434, A347438, A362610, A362611, A362612, A362614, A363723.
%K A363265 nonn
%O A363265 1,4
%A A363265 _Gus Wiseman_, Jun 27 2023