A362617 Numbers whose prime factorization has both (1) even length, and (2) unequal middle parts.

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159, 161, 166, 177

Offset: 1

Author

Gus Wiseman, May 10 2023

Keywords

• nonn

Comments

Also numbers n whose median prime factor is not a prime factor of n, where 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 prime factorization of 60 is 2*2*3*5, with middle parts (2,3), so 60 is in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Partitions of this type are counted by A238479.

The complement (without 1) is A362618, counted by A238478.

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

A359893 counts partitions by median.

A359908 ranks partitions with integer median, counted by A325347.

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

A362605 ranks partitions with more than one mode, counted by A362607.

A362611 counts modes in prime factorization, triangle version A362614.

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

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

Contains A006881 and (except for 1) A030229.

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

Programs

Maple

filter:= proc(n) local F,m;
  F:= sort(map(t -> t[1]$t[2],ifactors(n)[2]));
  m:= nops(F);
  m::even and F[m/2] <> F[m/2+1]
end proc:
select(filter, [$2..200]); # Robert Israel, Dec 15 2023

Mathematica

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