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A364158
Numbers whose multiset of prime factors has low (i.e. least) co-mode 2.
6
1, 2, 4, 6, 8, 10, 14, 16, 18, 22, 26, 30, 32, 34, 36, 38, 42, 46, 50, 54, 58, 62, 64, 66, 70, 74, 78, 82, 86, 90, 94, 98, 100, 102, 106, 108, 110, 114, 118, 122, 126, 128, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194
OFFSET
1,2
COMMENTS
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.
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}.
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.
Extending the terminology of A124943, the "low co-mode" of a multiset is the least co-mode.
EXAMPLE
The terms together with their prime factorizations begin:
1 =
2 = 2
4 = 2*2
6 = 2*3
8 = 2*2*2
10 = 2*5
14 = 2*7
16 = 2*2*2*2
18 = 2*3*3
22 = 2*11
26 = 2*13
30 = 2*3*5
32 = 2*2*2*2*2
34 = 2*17
36 = 2*2*3*3
MATHEMATICA
prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
comodes[ms_]:=Select[Union[ms], Count[ms, #]<=Min@@Length/@Split[ms]&];
Select[Range[100], #==1||Min[comodes[prifacs[#]]]==2&]
CROSSREFS
Partitions of this type are counted by A364159.
Positions of 1's in A364191, high A364192, modes A363486, high A363487.
For median we have A363488, positions of 1 in A363941, triangle A124943.
For mode instead of co-mode we have A360015, counted by A241131.
A027746 lists prime factors (with multiplicity), length A001222.
A362611 counts modes in prime factorization, triangle A362614
A362613 counts co-modes in prime factorization, triangle A362615
Ranking partitions:
- A356862: unique mode, counted by A362608
- A359178: unique co-mode, counted by A362610
- A362605: multiple modes, counted by A362607
- A362606: multiple co-modes, counted by A362609
Sequence in context: A022292 A225241 A087370 * A138929 A334169 A180081
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 14 2023
STATUS
approved