A360252 Numbers for which the prime indices have greater mean than the distinct prime indices.

18, 50, 54, 75, 98, 108, 147, 150, 162, 242, 245, 250, 294, 324, 338, 350, 363, 375, 450, 486, 490, 500, 507, 578, 588, 605, 648, 686, 722, 726, 735, 750, 845, 847, 867, 882, 972, 1014, 1029, 1050, 1058, 1078, 1083, 1125, 1183, 1210, 1250, 1274, 1350, 1372

Offset: 1

Author

Gus Wiseman, Feb 09 2023

Keywords

• nonn

Comments

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.

Examples

			The terms together with their prime indices begin:
    18: {1,2,2}
    50: {1,3,3}
    54: {1,2,2,2}
    75: {2,3,3}
    98: {1,4,4}
   108: {1,1,2,2,2}
   147: {2,4,4}
   150: {1,2,3,3}
   162: {1,2,2,2,2}
   242: {1,5,5}
   245: {3,4,4}
   250: {1,3,3,3}
   294: {1,2,4,4}
   324: {1,1,2,2,2,2}
For example, the prime indices of 350 are {1,3,3,4} with mean 11/4, and the distinct prime indices are {1,3,4} with mean 8/3, so 350 is in the sequence.
		

Crossrefs

For unequal instead of greater we have A360246, counted by A360242.

For equal instead of greater we have A360247, counted by A360243.

These partitions are counted by A360250.

For less instead of greater we have A360253, counted by A360251.

A112798 lists prime indices, length A001222, sum A056239.

A316413 lists numbers whose indices have integer mean, distinct A326621.

A326567/A326568 gives mean of prime indices.

A326619/A326620 gives mean of distinct prime indices.

Cf. A000975, A051293, A058398, A067340, A067538, A324570, A327482, A359903, A360005, A360241, A360248.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Mean[prix[#]]>Mean[Union[prix[#]]]&]