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Numbers whose prime indices have rounded-up mean 2.
12

%I #6 Jul 06 2023 08:56:01

%S 3,6,9,10,12,18,20,24,27,28,30,36,40,48,54,56,60,72,80,81,84,88,90,96,

%T 100,108,112,120,144,160,162,168,176,180,192,200,208,216,224,240,243,

%U 252,264,270,280,288,300,320,324,336,352,360,384,400,416,432,448

%N Numbers whose prime indices have rounded-up mean 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 terms together with their prime indices begin:

%e 3: {2}

%e 6: {1,2}

%e 9: {2,2}

%e 10: {1,3}

%e 12: {1,1,2}

%e 18: {1,2,2}

%e 20: {1,1,3}

%e 24: {1,1,1,2}

%e 27: {2,2,2}

%e 28: {1,1,4}

%e 30: {1,2,3}

%e 36: {1,1,2,2}

%e 40: {1,1,1,3}

%e 48: {1,1,1,1,2}

%e 54: {1,2,2,2}

%e 56: {1,1,1,4}

%e 60: {1,1,2,3}

%e 72: {1,1,1,2,2}

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

%e 81: {2,2,2,2}

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

%t Select[Range[1000],Ceiling[Mean[prix[#]]]==2&]

%Y For mean 1 we have A000079 except 1.

%Y Partitions of this type are counted by A026905 redoubled.

%Y Equals the complement of A000079 in A344296.

%Y Positions of 2's in A363944 (counted by column 2 of A363946).

%Y For rounded mean 1 we have A363948, counted by A363947.

%Y For rounded-down mean 1 we have A363949, counted by A025065.

%Y The rounded-down or low version is A363954, counted by A363745.

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

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

%Y A326567/A326568 gives mean of prime indices.

%Y A363941 gives low median of prime indices, triangle A124943.

%Y A363942 gives high median of prime indices, triangle A124944.

%Y Cf. A327473, A327476, A359889, A360005, A360013, A360015, A363727, A363943.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 05 2023