A360459 Two times the median of the multiset of prime factors of n; a(1) = 2.

2, 4, 6, 4, 10, 5, 14, 4, 6, 7, 22, 4, 26, 9, 8, 4, 34, 6, 38, 4, 10, 13, 46, 4, 10, 15, 6, 4, 58, 6, 62, 4, 14, 19, 12, 5, 74, 21, 16, 4, 82, 6, 86, 4, 6, 25, 94, 4, 14, 10, 20, 4, 106, 6, 16, 4, 22, 31, 118, 5, 122, 33, 6, 4, 18, 6, 134, 4, 26, 10, 142, 4, 146

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.

Examples

			The prime factors of 60 are {2,2,3,5}, with median 5/2, so a(60) = 5.
		

Crossrefs

The union is 2 followed by A014091, complement of A014092.

The prime factors themselves are listed by A027746, distinct A027748.

The version for divisors is A063655.

Positions of odd terms are A072978 (except 1).

For mean instead of twice median: A123528/A123529, distinct A323171/A323172.

Positions of even terms are A359913 (and 1).

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 multiplicities is A360460.

The version for 0-prepended differences is A360555.

A112798 lists prime indices, length A001222, sum A056239.

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, A027336, A078174, A316413, A359907, A359908, A360006, A360007, A360248, A360552.

Programs

Mathematica

Table[2*Median[Join@@ConstantArray@@@FactorInteger[n]],{n,100}]