A363730 Numbers whose prime indices have different mean, median, and mode.

42, 60, 66, 70, 78, 84, 102, 114, 130, 132, 138, 140, 150, 154, 156, 165, 170, 174, 180, 182, 186, 190, 195, 204, 220, 222, 228, 230, 231, 246, 255, 258, 260, 266, 276, 282, 285, 286, 290, 294, 308, 310, 315, 318, 322, 330, 340, 345, 348, 354, 357, 360, 364

Offset: 1

Author

Gus Wiseman, Jun 24 2023

Keywords

• nonn

Comments

If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.

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.

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The prime indices of 180 are {1,1,2,2,3}, with mean 9/5, median 2, modes {1,2}, so 180 is in the sequence.
The prime indices of 108 are {1,1,2,2,2}, with mean 8/5, median 2, modes {2}, so 108 is not in the sequence.
The terms together with their prime indices begin:
   42: {1,2,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  132: {1,1,2,5}
  138: {1,2,9}
  140: {1,1,3,4}
  150: {1,2,3,3}
		

Crossrefs

These partitions are counted by A363720

For equal instead of unequal we have A363727, counted by A363719.

The version for factorizations is A363742, equal A363741.

A112798 lists prime indices, length A001222, sum A056239.

A326567/A326568 gives mean of prime indices.

A356862 ranks partitions with a unique mode, counted by A362608.

A359178 ranks partitions with multiple modes, counted by A362610.

A360005 gives twice the median of prime indices.

A362611 counts modes in prime indices, triangle A362614.

A362613 counts co-modes in prime indices, triangle A362615.

A363486 gives least mode in prime indices, A363487 greatest.

Just two statistics:

- (mean) = (median): A359889, counted by A240219.

- (mean) != (median): A359890, counted by A359894.

- (mean) = (mode): counted by A363723, see A363724, A363731.

- (median) = (mode): counted by A363740.

Cf. A000961, A327473, A327476, A359908, A363722, A363725, A363729.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[100],{Mean[prix[#]]}!={Median[prix[#]]}!=modes[prix[#]]&]

Formula

All three of A326567(a(n))/A326568(a(n)), A360005(a(n))/2, and A363486(a(n)) = A363487(a(n)) are different.