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A364056 Numbers whose prime factors have high median 2. Numbers whose prime factors (with multiplicity) are mostly 2's.

%I A364056 #7 Jul 08 2023 08:04:36
%S A364056 2,4,8,12,16,20,24,28,32,40,44,48,52,56,64,68,72,76,80,88,92,96,104,
%T A364056 112,116,120,124,128,136,144,148,152,160,164,168,172,176,184,188,192,
%U A364056 200,208,212,224,232,236,240,244,248,256,264,268,272,280,284,288,292
%N A364056 Numbers whose prime factors have high median 2. Numbers whose prime factors (with multiplicity) are mostly 2's.
%C A364056 The multiset of prime factors of n is row n of A027746.
%C A364056 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 A364056 The terms together with their prime indices begin:
%e A364056      2: {1}             64: {1,1,1,1,1,1}      136: {1,1,1,7}
%e A364056      4: {1,1}           68: {1,1,7}            144: {1,1,1,1,2,2}
%e A364056      8: {1,1,1}         72: {1,1,1,2,2}        148: {1,1,12}
%e A364056     12: {1,1,2}         76: {1,1,8}            152: {1,1,1,8}
%e A364056     16: {1,1,1,1}       80: {1,1,1,1,3}        160: {1,1,1,1,1,3}
%e A364056     20: {1,1,3}         88: {1,1,1,5}          164: {1,1,13}
%e A364056     24: {1,1,1,2}       92: {1,1,9}            168: {1,1,1,2,4}
%e A364056     28: {1,1,4}         96: {1,1,1,1,1,2}      172: {1,1,14}
%e A364056     32: {1,1,1,1,1}    104: {1,1,1,6}          176: {1,1,1,1,5}
%e A364056     40: {1,1,1,3}      112: {1,1,1,1,4}        184: {1,1,1,9}
%e A364056     44: {1,1,5}        116: {1,1,10}           188: {1,1,15}
%e A364056     48: {1,1,1,1,2}    120: {1,1,1,2,3}        192: {1,1,1,1,1,1,2}
%e A364056     52: {1,1,6}        124: {1,1,11}           200: {1,1,1,3,3}
%e A364056     56: {1,1,1,4}      128: {1,1,1,1,1,1,1}    208: {1,1,1,1,6}
%t A364056 prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t A364056 merr[y_]:=If[Length[y]==0,0,If[OddQ[Length[y]],y[[(Length[y]+1)/2]], y[[1+Length[y]/2]]]];
%t A364056 Select[Range[100],merr[prifacs[#]]==2&]
%Y A364056 Partitions of this type are counted by A027336.
%Y A364056 Median of prime indices is A360005(n)/2.
%Y A364056 For mode instead of median we have A360013, low A360015.
%Y A364056 The low version is A363488, positions of 1's in A363941.
%Y A364056 Positions of 1's in A363942.
%Y A364056 A112798 lists prime indices, length A001222, sum A056239.
%Y A364056 A123528/A123529 gives mean of prime factors, indices A326567/A326568.
%Y A364056 A124943 counts partitions by low median, high A124944.
%Y A364056 Cf. A072978, A215366, A316413, A359908, A363727, A363740, A363949.
%K A364056 nonn
%O A364056 1,1
%A A364056 _Gus Wiseman_, Jul 07 2023