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

%I A362617 #10 Dec 16 2023 09:00:30
%S A362617 6,10,14,15,21,22,26,33,34,35,36,38,39,46,51,55,57,58,60,62,65,69,74,
%T A362617 77,82,84,85,86,87,91,93,94,95,100,106,111,115,118,119,122,123,129,
%U A362617 132,133,134,140,141,142,143,145,146,150,155,156,158,159,161,166,177
%N A362617 Numbers whose prime factorization has both (1) even length, and (2) unequal middle parts.
%C A362617 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 A362617 Robert Israel, <a href="/A362617/b362617.txt">Table of n, a(n) for n = 1..10000</a>
%e A362617 The prime factorization of 60 is 2*2*3*5, with middle parts (2,3), so 60 is in the sequence.
%p A362617 filter:= proc(n) local F,m;
%p A362617   F:= sort(map(t -> t[1]$t[2],ifactors(n)[2]));
%p A362617   m:= nops(F);
%p A362617   m::even and F[m/2] <> F[m/2+1]
%p A362617 end proc:
%p A362617 select(filter, [$2..200]); # _Robert Israel_, Dec 15 2023
%t A362617 prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t A362617 Select[Range[2,100],FreeQ[prifacs[#],Median[prifacs[#]]]&]
%Y A362617 Partitions of this type are counted by A238479.
%Y A362617 The complement (without 1) is A362618, counted by A238478.
%Y A362617 A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
%Y A362617 A359893 counts partitions by median.
%Y A362617 A359908 ranks partitions with integer median, counted by A325347.
%Y A362617 A359912 ranks partitions with non-integer median, counted by A307683.
%Y A362617 A362605 ranks partitions with more than one mode, counted by A362607.
%Y A362617 A362611 counts modes in prime factorization, triangle version A362614.
%Y A362617 A362621 ranks partitions with median equal to maximum, counted by A053263.
%Y A362617 A362622 ranks partitions whose maximum is a middle part, counted by A237824.
%Y A362617 Contains A006881 and (except for 1) A030229.
%Y A362617 Cf. A000040, A171979, A327473, A327476, A356862, A359907, A362616, A362620.
%K A362617 nonn
%O A362617 1,1
%A A362617 _Gus Wiseman_, May 10 2023