A362617 - OEIS

%I #10 Dec 16 2023 09:00:30

%S 6,10,14,15,21,22,26,33,34,35,36,38,39,46,51,55,57,58,60,62,65,69,74,

%T 77,82,84,85,86,87,91,93,94,95,100,106,111,115,118,119,122,123,129,

%U 132,133,134,140,141,142,143,145,146,150,155,156,158,159,161,166,177

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

%C 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).

%H Robert Israel, <a href="/A362617/b362617.txt">Table of n, a(n) for n = 1..10000</a>

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

%p filter:= proc(n) local F,m;

%p   F:= sort(map(t -> t[1]$t[2],ifactors(n)[2]));

%p   m:= nops(F);

%p   m::even and F[m/2] <> F[m/2+1]

%p end proc:

%p select(filter, [$2..200]); # _Robert Israel_, Dec 15 2023

%t prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];

%t Select[Range[2,100],FreeQ[prifacs[#],Median[prifacs[#]]]&]

%Y Partitions of this type are counted by A238479.

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

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

%Y A359893 counts partitions by median.

%Y A359908 ranks partitions with integer median, counted by A325347.

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

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

%Y A362611 counts modes in prime factorization, triangle version A362614.

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

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

%Y Contains A006881 and (except for 1) A030229.

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

%K nonn

%O 1,1

%A _Gus Wiseman_, May 10 2023