A362622 One and numbers whose prime factorization has its greatest part at a middle position.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91

Offset: 1

Author

Gus Wiseman, May 12 2023

Keywords

• nonn

Examples

			The prime factorization of 150 is 5*5*3*2, with middle parts {3,5}, so 150 is in the sequence.
The prime factorization of 90 is 5*3*3*2, with middle parts {3,3}, so 90 is not in the sequence.
		

Crossrefs

Partitions of this type are counted by A237824.

For modes instead of middles we have A362619, counted by A171979.

The version for median instead of middles is A362621, counted by A053263.

The complement for median is A362980, counted by A237821.

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

A362611 counts modes in prime factorization.

A362613 counts co-modes in prime factorization.

Cf. A000040, A002865, A237984, A327473, A327476, A359908, A362616, A362620.

Programs

Mathematica

mpm[q_]:=MemberQ[If[OddQ[Length[q]],{Median[q]},{q[[Length[q]/2]],q[[Length[q]/2+1]]}],Max@@q];
Select[Range[100],#==1||mpm[Flatten[Apply[ConstantArray,FactorInteger[#],{1}]]]&]