A363948 - OEIS

%I #14 Jul 09 2023 16:27:01

%S 2,4,8,12,16,24,32,48,64,72,80,96,128,144,160,192,256,288,320,384,432,

%T 448,480,512,576,640,768,864,896,960,1024,1152,1280,1536,1728,1792,

%U 1920,2048,2304,2560,2592,2688,2816,2880,3072,3200,3456,3584,3840,4096,4608

%N Numbers whose prime indices have mean < 3/2.

%C 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.

%e The initial terms, prime indices, and means:

%e     2: {1} -> 1

%e     4: {1,1} -> 1

%e     8: {1,1,1} -> 1

%e    12: {1,1,2} -> 4/3

%e    16: {1,1,1,1} -> 1

%e    24: {1,1,1,2} -> 5/4

%e    32: {1,1,1,1,1} -> 1

%e    48: {1,1,1,1,2} -> 6/5

%e    64: {1,1,1,1,1,1} -> 1

%e    72: {1,1,1,2,2} -> 7/5

%e    80: {1,1,1,1,3} -> 7/5

%e    96: {1,1,1,1,1,2} -> 7/6

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Mean[prix[#]]<3/2&]

%Y These partitions are counted by A363947.

%Y Prime indices have mean A326567/A326568.

%Y For low mode we have A360015, high A360013.

%Y Positions of 1's in A363489.

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

%Y A316413 ranks partitions with integer mean, counted by A067538.

%Y A360005 gives twice the median of prime indices.

%Y A363949 ranks partitions with low mean 1, counted by A025065.

%Y A363950 ranks partitions with low mean 2, counted by A026905 redoubled.

%Y Cf. A051293, A124944, A327473, A327476, A327482, A359889, A363727, A363942, A363943, A363946, A363951.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 02 2023