|
|
A363730
|
|
Numbers whose prime indices have different mean, median, and mode.
|
|
10
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
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).
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
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}
|
|
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[#]]&]
|
|
CROSSREFS
|
These partitions are counted by A363720
A360005 gives twice the median of prime indices.
Just two statistics:
- (median) = (mode): counted by A363740.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|