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

6, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 104, 105, 106, 110

Offset: 1

Author

Gus Wiseman, May 12 2023

Keywords

• nonn

Comments

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

Examples

			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.
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.
The terms together with their prime indices begin:
     6: {1,2}        36: {1,1,2,2}      60: {1,1,2,3}
    10: {1,3}        38: {1,8}          62: {1,11}
    12: {1,1,2}      39: {2,6}          63: {2,2,4}
    14: {1,4}        40: {1,1,1,3}      65: {3,6}
    15: {2,3}        42: {1,2,4}        66: {1,2,5}
    20: {1,1,3}      44: {1,1,5}        68: {1,1,7}
    21: {2,4}        45: {2,2,3}        69: {2,9}
    22: {1,5}        46: {1,9}          70: {1,3,4}
    24: {1,1,1,2}    48: {1,1,1,1,2}    72: {1,1,1,2,2}
    26: {1,6}        51: {2,7}          74: {1,12}
    28: {1,1,4}      52: {1,1,6}        76: {1,1,8}
    30: {1,2,3}      55: {3,5}          77: {4,5}
    33: {2,5}        56: {1,1,1,4}      78: {1,2,6}
    34: {1,7}        57: {2,8}          80: {1,1,1,1,3}
    35: {3,4}        58: {1,10}         82: {1,13}
		

Crossrefs

Partitions of this type are counted by A237821.

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

The complement is A362621, counted by A053263.

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

A362611 counts modes in prime factorization, triangle version A362614.

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

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

Programs

Mathematica

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