A360457 Two times the median of the set of distinct prime indices of n; a(1) = 1.

1, 2, 4, 2, 6, 3, 8, 2, 4, 4, 10, 3, 12, 5, 5, 2, 14, 3, 16, 4, 6, 6, 18, 3, 6, 7, 4, 5, 20, 4, 22, 2, 7, 8, 7, 3, 24, 9, 8, 4, 26, 4, 28, 6, 5, 10, 30, 3, 8, 4, 9, 7, 32, 3, 8, 5, 10, 11, 34, 4, 36, 12, 6, 2, 9, 4, 38, 8, 11, 6, 40, 3, 42, 13, 5, 9, 9, 4, 44, 4

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 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. Distinct prime indices are listed by A304038.

Examples

			The prime indices of 65 are {3,6}, with distinct parts {3,6}, with median 9/2, so a(65) = 9.
The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so a(900) = 4.
		

Crossrefs

The version for divisors is A063655.

For mean instead of two times median we have A326619/A326620.

The version for all prime indices is A360005.

Positions of first appearances are A360006, sorted A360007.

The version for distinct prime factors is A360458.

The version for all prime factors is A360459.

The version for prime multiplicities is A360460.

Positions of even terms are A360550.

Positions of odd terms are A360551.

The version for 0-prepended differences is A360555.

A112798 lists prime indices, length A001222, sum A056239.

A304038 lists distinct prime indices.

A325347 counts partitions with integer median, complement A307683.

A326567/A326568 gives mean of prime indices.

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

Cf. A000975, A026424, A316413, A359907, A359908, A359912, A360248, A360453.

Programs

Mathematica

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