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

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

Author

Gus Wiseman, Jul 14 2023

Keywords

• nonn

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.

Examples

			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
		

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

Cf. A215366, A327473, A327476, A360005.

Programs

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&]