A364056 - OEIS

%I #7 Jul 08 2023 08:04:36

%S 2,4,8,12,16,20,24,28,32,40,44,48,52,56,64,68,72,76,80,88,92,96,104,

%T 112,116,120,124,128,136,144,148,152,160,164,168,172,176,184,188,192,

%U 200,208,212,224,232,236,240,244,248,256,264,268,272,280,284,288,292

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

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

%C 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).

%e The terms together with their prime indices begin:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%Y Partitions of this type are counted by A027336.

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

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

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

%Y Positions of 1's in A363942.

%Y A112798 lists prime indices, length A001222, sum A056239.

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

%Y A124943 counts partitions by low median, high A124944.

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 07 2023