%I #6 May 12 2023 12:43:45

%S 6,10,12,14,15,20,21,22,24,26,28,30,33,34,35,36,38,39,40,42,44,45,46,

%T 48,51,52,55,56,57,58,60,62,63,65,66,68,69,70,72,74,76,77,78,80,82,84,

%U 85,86,87,88,90,91,92,93,94,95,96,99,100,102,104,105,106,110

%N Numbers whose multiset of prime factors (with multiplicity) has different median from maximum.

%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

%e The prime factorization of 108 is 2*2*3*3*3, and the multiset {2,2,3,3,3} has median 3 and maximum 3, so 108 is not in the sequence.

%e The prime factorization of 2250 is 2*3*3*5*5*5, and the multiset {2,3,3,5,5,5} has median 4 and maximum 5, so 2250 is in the sequence.

%e The terms together with their prime indices begin:

%e 6: {1,2} 36: {1,1,2,2} 60: {1,1,2,3}

%e 10: {1,3} 38: {1,8} 62: {1,11}

%e 12: {1,1,2} 39: {2,6} 63: {2,2,4}

%e 14: {1,4} 40: {1,1,1,3} 65: {3,6}

%e 15: {2,3} 42: {1,2,4} 66: {1,2,5}

%e 20: {1,1,3} 44: {1,1,5} 68: {1,1,7}

%e 21: {2,4} 45: {2,2,3} 69: {2,9}

%e 22: {1,5} 46: {1,9} 70: {1,3,4}

%e 24: {1,1,1,2} 48: {1,1,1,1,2} 72: {1,1,1,2,2}

%e 26: {1,6} 51: {2,7} 74: {1,12}

%e 28: {1,1,4} 52: {1,1,6} 76: {1,1,8}

%e 30: {1,2,3} 55: {3,5} 77: {4,5}

%e 33: {2,5} 56: {1,1,1,4} 78: {1,2,6}

%e 34: {1,7} 57: {2,8} 80: {1,1,1,1,3}

%e 35: {3,4} 58: {1,10} 82: {1,13}

%t Select[Range[100],(y=Flatten[Apply[ConstantArray,FactorInteger[#],{1}]];Max@@y!=Median[y])&]

%Y Partitions of this type are counted by A237821.

%Y For mode instead of median we have A362620, counted by A240302.

%Y The complement is A362621, counted by A053263.

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

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

%Y A362613 counts co-modes in prime factorization, triangle version A362615.

%Y Cf. A000040, A002865, A171979, A237824, A237984, A327473, A327476, A359908, A362616, A362619, A362622.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 12 2023