A364056 - OEIS

A364056

Numbers whose prime factors have high median 2. Numbers whose prime factors (with multiplicity) are mostly 2's.

2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 64, 68, 72, 76, 80, 88, 92, 96, 104, 112, 116, 120, 124, 128, 136, 144, 148, 152, 160, 164, 168, 172, 176, 184, 188, 192, 200, 208, 212, 224, 232, 236, 240, 244, 248, 256, 264, 268, 272, 280, 284, 288, 292

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OFFSET

1,1

COMMENTS

The multiset of prime factors of n is row n of A027746.

The high median (see A124944) in a multiset is either the middle part (for odd length), or the greatest of the two middle parts (for even length).

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

The terms together with their prime indices begin:

2: {1} 64: {1,1,1,1,1,1} 136: {1,1,1,7}

4: {1,1} 68: {1,1,7} 144: {1,1,1,1,2,2}

8: {1,1,1} 72: {1,1,1,2,2} 148: {1,1,12}

12: {1,1,2} 76: {1,1,8} 152: {1,1,1,8}

16: {1,1,1,1} 80: {1,1,1,1,3} 160: {1,1,1,1,1,3}

20: {1,1,3} 88: {1,1,1,5} 164: {1,1,13}

24: {1,1,1,2} 92: {1,1,9} 168: {1,1,1,2,4}

28: {1,1,4} 96: {1,1,1,1,1,2} 172: {1,1,14}

32: {1,1,1,1,1} 104: {1,1,1,6} 176: {1,1,1,1,5}

40: {1,1,1,3} 112: {1,1,1,1,4} 184: {1,1,1,9}

44: {1,1,5} 116: {1,1,10} 188: {1,1,15}

48: {1,1,1,1,2} 120: {1,1,1,2,3} 192: {1,1,1,1,1,1,2}

52: {1,1,6} 124: {1,1,11} 200: {1,1,1,3,3}

56: {1,1,1,4} 128: {1,1,1,1,1,1,1} 208: {1,1,1,1,6}

MATHEMATICA

prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];

merr[y_]:=If[Length[y]==0, 0, If[OddQ[Length[y]], y[[(Length[y]+1)/2]], y[[1+Length[y]/2]]]];

Select[Range[100], merr[prifacs[#]]==2&]

CROSSREFS

Partitions of this type are counted by A027336.

Median of prime indices is A360005(n)/2.

For mode instead of median we have A360013, low A360015.

The low version is A363488, positions of 1's in A363941.

Positions of 1's in A363942.

A112798 lists prime indices, length A001222, sum A056239.

A123528/A123529 gives mean of prime factors, indices A326567/A326568.

A124943 counts partitions by low median, high A124944.

Cf. A072978, A215366, A316413, A359908, A363727, A363740, A363949.

Sequence in context: A048166 A360013 A335738 * A010066 A180490 A376616

Adjacent sequences: A364053 A364054 A364055 * A364057 A364058 A364059

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 07 2023

STATUS

approved