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A362980 Numbers whose multiset of prime factors (with multiplicity) has different median from maximum.

%I A362980 #6 May 12 2023 12:43:45
%S A362980 6,10,12,14,15,20,21,22,24,26,28,30,33,34,35,36,38,39,40,42,44,45,46,
%T A362980 48,51,52,55,56,57,58,60,62,63,65,66,68,69,70,72,74,76,77,78,80,82,84,
%U A362980 85,86,87,88,90,91,92,93,94,95,96,99,100,102,104,105,106,110
%N A362980 Numbers whose multiset of prime factors (with multiplicity) has different median from maximum.
%C A362980 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 A362980 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 A362980 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 A362980 The terms together with their prime indices begin:
%e A362980      6: {1,2}        36: {1,1,2,2}      60: {1,1,2,3}
%e A362980     10: {1,3}        38: {1,8}          62: {1,11}
%e A362980     12: {1,1,2}      39: {2,6}          63: {2,2,4}
%e A362980     14: {1,4}        40: {1,1,1,3}      65: {3,6}
%e A362980     15: {2,3}        42: {1,2,4}        66: {1,2,5}
%e A362980     20: {1,1,3}      44: {1,1,5}        68: {1,1,7}
%e A362980     21: {2,4}        45: {2,2,3}        69: {2,9}
%e A362980     22: {1,5}        46: {1,9}          70: {1,3,4}
%e A362980     24: {1,1,1,2}    48: {1,1,1,1,2}    72: {1,1,1,2,2}
%e A362980     26: {1,6}        51: {2,7}          74: {1,12}
%e A362980     28: {1,1,4}      52: {1,1,6}        76: {1,1,8}
%e A362980     30: {1,2,3}      55: {3,5}          77: {4,5}
%e A362980     33: {2,5}        56: {1,1,1,4}      78: {1,2,6}
%e A362980     34: {1,7}        57: {2,8}          80: {1,1,1,1,3}
%e A362980     35: {3,4}        58: {1,10}         82: {1,13}
%t A362980 Select[Range[100],(y=Flatten[Apply[ConstantArray,FactorInteger[#],{1}]];Max@@y!=Median[y])&]
%Y A362980 Partitions of this type are counted by A237821.
%Y A362980 For mode instead of median we have A362620, counted by A240302.
%Y A362980 The complement is A362621, counted by A053263.
%Y A362980 A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
%Y A362980 A362611 counts modes in prime factorization, triangle version A362614.
%Y A362980 A362613 counts co-modes in prime factorization, triangle version A362615.
%Y A362980 Cf. A000040, A002865, A171979, A237824, A237984, A327473, A327476, A359908, A362616, A362619, A362622.
%K A362980 nonn
%O A362980 1,1
%A A362980 _Gus Wiseman_, May 12 2023