cp's OEIS Frontend

A363741 Number of factorizations of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

%I A363741 #14 Aug 16 2023 11:05:38
%S A363741 0,1,1,2,1,1,1,2,2,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,2,1,1,
%T A363741 1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,
%U A363741 1,1,1,2,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1
%N A363741 Number of factorizations of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.
%C A363741 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 A363741 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%C A363741 Position of first appearance of n is: (1, 2, 4, 16, 64, 5832, 4096, ...).
%e A363741 The factorization 6*9*9*12 = 5832 has mean 9, median 9, and modes {9}, so it is counted under a(5832).
%e A363741 The a(n) factorizations for selected n:
%e A363741 2   4     16        64            5832              4096
%e A363741     2*2   4*4       8*8           18*18*18          64*64
%e A363741           2*2*2*2   4*4*4         6*9*9*12          8*8*8*8
%e A363741                     2*2*2*2*2*2   3*6*6*6*9         16*16*16
%e A363741                                   2*3*3*3*3*3*3*4   4*4*4*4*4*4
%e A363741                                                     2*2*2*2*2*2*2*2*2*2*2*2
%t A363741 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A363741 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A363741 Table[Length[Select[facs[n],{Mean[#]}=={Median[#]}==modes[#]&]],{n,100}]
%Y A363741 For just (mean) = (median): A359909, see A240219, A359889, A359910, A359911.
%Y A363741 The version for partitions is A363719, unequal A363720.
%Y A363741 For unequal instead of equal we have A363742.
%Y A363741 A000041 counts integer partitions.
%Y A363741 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A363741 A089723 counts constant factorizations.
%Y A363741 A316439 counts factorizations by length, A008284 partitions.
%Y A363741 A326622 counts factorizations with integer mean, strict A328966.
%Y A363741 A339846 counts even-length factorizations, A339890 odd-length.
%Y A363741 Cf. A237984, A326567/A326568, A327472, A359893, A363723, A363727, A363740.
%K A363741 nonn
%O A363741 1,4
%A A363741 _Gus Wiseman_, Jun 26 2023