A360553 Numbers > 1 whose unordered prime signature has integer median.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83

Offset: 1

Author

Gus Wiseman, Feb 16 2023

Keywords

• nonn

Comments

First differs from A067340 in having 60.

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

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 unordered prime signature of 60 is {1,1,2}, with median 1, so 60 is in the sequence.
The unordered prime signature of 1260 is {1,1,2,2}, with median 3/2, so 1260 is not in the sequence.
		

Crossrefs

For mean instead of median we have A067340, complement A070011.

Positions of even terms in A360460.

The complement is A360554 (without 1).

These partitions are counted by A360687.

- For divisors (A063655) we have A139711, complement A139710.

- For prime indices (A360005) we have A359908, complement A359912.

- For distinct prime indices (A360457) we have A360550, complement A360551.

- For distinct prime factors (A360458) we have A360552, complement A100367.

- For prime factors (A360459) we have A359913, complement A072978.

- For prime multiplicities (A360460) we have A360553, complement A360554.

- For 0-prepended differences (A360555) we have A360556, complement A360557.

A112798 lists prime indices, length A001222, sum A056239.

A124010 lists prime signature.

A325347 = partitions w/ integer median, complement A307683, strict A359907.

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

A360454 = numbers whose prime indices and signature have the same median.

Cf. A000975, A026424, A078174, A078175, A316413, A326621, A360453.

Programs

Mathematica

Select[Range[2,100],IntegerQ[Median[Last/@FactorInteger[#]]]&]