A362618 Numbers whose prime factorization has either (1) odd length, or (2) equal middle parts.

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 88, 89, 90, 92, 96, 97, 98, 99, 101

Offset: 1

Author

Gus Wiseman, May 10 2023

Keywords

• nonn

Comments

Also numbers n whose median prime factor is a prime factor of n.

Examples

			The prime factorization of 90 is 2*3*3*5, with middle parts (3,3), so 90 is in the sequence.
		

Crossrefs

Partitions of this type are counted by A238478.

The complement (without 1) is A362617, counted by A238479.

A027746 lists prime factors, A112798 indices, length A001222, sum A056239.

A359178 ranks partitions with a unique co-mode, counted by A362610.

A359893 counts partitions by median.

A359908 ranks partitions with integer median, counted by A325347.

A359912 ranks partitions with non-integer median, counted by A307683.

A362611 ranks modes in prime factorization, counted by A362614.

A362621 ranks partitions with median equal to maximum, counted by A053263.

A362622 ranks partitions whose maximum is a middle part, counted by A237824.

Cf. A000040, A171979, A327473, A327476, A356862, A359907, A360686, A360952, A362616, A362620.

Programs

Mathematica

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2,100],MemberQ[prifacs[#],Median[prifacs[#]]]&]