A359889 Numbers that are 1 or whose prime indices have the same mean as median.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94

Offset: 1

Author

Gus Wiseman, Jan 22 2023

Keywords

• nonn

Comments

First differs from A236510 in having 252 (prime indices {1,1,2,2,4}).

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).

Examples

			The prime indices of 900 are {1,1,2,2,3,3}, with mean 2 and median 2, so 900 is in the sequence.
		

Crossrefs

These partitions are counted by A240219, strict A359897.

The LHS (mean of prime indices) is A326567/A326568.

The complement is A359890, counted by A359894.

The odd-length case is A359891, complement A359892, counted by A359895.

The RHS (median of prime indices) is A360005/2.

A058398 counts partitions by mean, see also A008284, A327482.

A088529/A088530 gives mean of prime signature A124010.

A112798 lists prime indices, length A001222, sum A056239.

A316413 lists numbers whose prime indices have integer mean.

A359893 and A359901 count partitions by median, odd-length A359902.

A359908 lists numbers whose prime indices have integer median.

Cf. A026424, A327473, A359899, A359903, A359905, A359909, A360006, A360007, A360008, A360009.

Programs

Mathematica

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

Formula

Numbers n such that A326567(n)/A326568(n) = A360005(n)/2.