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A364158 Numbers whose multiset of prime factors has low (i.e. least) co-mode 2.

%I A364158 #15 Oct 18 2023 04:46:38
%S A364158 1,2,4,6,8,10,14,16,18,22,26,30,32,34,36,38,42,46,50,54,58,62,64,66,
%T A364158 70,74,78,82,86,90,94,98,100,102,106,108,110,114,118,122,126,128,130,
%U A364158 134,138,142,146,150,154,158,162,166,170,174,178,182,186,190,194
%N A364158 Numbers whose multiset of prime factors has low (i.e. least) co-mode 2.
%C A364158 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A364158 We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.
%C A364158 Except for 1, this is the lists of all even numbers whose prime factorization contains at most as many 2's as non-2 parts.
%C A364158 Extending the terminology of A124943, the "low co-mode" of a multiset is the least co-mode.
%e A364158 The terms together with their prime factorizations begin:
%e A364158    1 =
%e A364158    2 = 2
%e A364158    4 = 2*2
%e A364158    6 = 2*3
%e A364158    8 = 2*2*2
%e A364158   10 = 2*5
%e A364158   14 = 2*7
%e A364158   16 = 2*2*2*2
%e A364158   18 = 2*3*3
%e A364158   22 = 2*11
%e A364158   26 = 2*13
%e A364158   30 = 2*3*5
%e A364158   32 = 2*2*2*2*2
%e A364158   34 = 2*17
%e A364158   36 = 2*2*3*3
%t A364158 prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t A364158 comodes[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
%t A364158 Select[Range[100],#==1||Min[comodes[prifacs[#]]]==2&]
%Y A364158 Partitions of this type are counted by A364159.
%Y A364158 Positions of 1's in A364191, high A364192, modes A363486, high A363487.
%Y A364158 For median we have A363488, positions of 1 in A363941, triangle A124943.
%Y A364158 For mode instead of co-mode we have A360015, counted by A241131.
%Y A364158 A027746 lists prime factors (with multiplicity), length A001222.
%Y A364158 A362611 counts modes in prime factorization, triangle A362614
%Y A364158 A362613 counts co-modes in prime factorization, triangle A362615
%Y A364158 Ranking partitions:
%Y A364158 - A356862: unique mode, counted by A362608
%Y A364158 - A359178: unique co-mode, counted by A362610
%Y A364158 - A362605: multiple modes, counted by A362607
%Y A364158 - A362606: multiple co-modes, counted by A362609
%Y A364158 Cf. A215366, A327473, A327476, A360005.
%K A364158 nonn
%O A364158 1,2
%A A364158 _Gus Wiseman_, Jul 14 2023