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

1,1

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).

Links

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

Example

The prime factorization of 60 is 2*2*3*5, with middle parts (2,3), so 60 is in the sequence.

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[#]]]&]

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.

Sequence in context: A325259 A320911 A371781 * A374472 A238748 A367590

Adjacent sequences: A362614 A362615 A362616 * A362618 A362619 A362620

Keyword

nonn

Author

Gus Wiseman, May 10 2023

Status

approved