A360460 Two times the median of the unordered prime signature of n; a(1) = 1.

1, 2, 2, 4, 2, 2, 2, 6, 4, 2, 2, 3, 2, 2, 2, 8, 2, 3, 2, 3, 2, 2, 2, 4, 4, 2, 6, 3, 2, 2, 2, 10, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 5, 4, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 2, 2, 2, 3, 12, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 5, 8, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Feb 14 2023

Keywords

• nonn

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

Examples

			The unordered prime signature of 2520 is {1,1,2,3}, with median 3/2, so a(2520) = 3.
		

Crossrefs

The version for divisors is A063655.

For mean instead of two times median we have A088529/A088530.

Prime signature is A124010, unordered A118914.

The version for prime indices is A360005.

The version for distinct prime indices is A360457.

The version for distinct prime factors is A360458.

The version for prime factors is A360459.

Positions of even terms are A360553.

Positions of odd terms are A360554.

The version for 0-prepended differences is A360555.

A112798 lists prime indices, length A001222, sum A056239.

A304038 lists distinct prime indices.

A325347 counts partitions w/ integer median, complement A307683.

A329976 counts partitions with median multiplicity 1.

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

Cf. A000975, A026424, A133464, A359907, A359908, A360454.

Programs

Mathematica

Table[If[n==1,1,2*Median[Last/@FactorInteger[n]]],{n,100}]