|
|
A360454
|
|
Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.
|
|
9
|
|
|
1, 2, 9, 54, 100, 120, 125, 135, 168, 180, 189, 240, 252, 264, 280, 297, 300, 312, 336, 351, 396, 408, 440, 450, 456, 459, 468, 480, 513, 520, 528, 540, 552, 560, 588, 612, 616, 621, 624, 672, 680, 684, 696, 728, 744, 756, 760, 783, 816, 828, 837, 880, 882
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
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.
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
|
|
|
EXAMPLE
|
The terms together with their prime indices begin:
1: {}
2: {1}
9: {2,2}
54: {1,2,2,2}
100: {1,1,3,3}
120: {1,1,1,2,3}
125: {3,3,3}
135: {2,2,2,3}
168: {1,1,1,2,4}
180: {1,1,2,2,3}
189: {2,2,2,4}
240: {1,1,1,1,2,3}
For example, the prime indices of 336 are {1,1,1,1,2,4} with median 1 and multiplicities {1,1,4} with median 1, so 336 is in the sequence.
|
|
MATHEMATICA
|
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Median[prix[#]]==Median[Length/@Split[prix[#]]]&]
|
|
CROSSREFS
|
For distinct indices instead of indices we have A360453, counted by A360455.
For distinct indices instead of multiplicities: A360249, counted by A360245.
These partitions are counted by A360456.
A240219 counts partitions with mean equal to median, ranked by A359889.
A359894 counts partitions with mean different from median, ranks A359890.
A360005 gives median of prime indices (times two).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|