A362621 One and numbers whose multiset of prime factors (with multiplicity) has the same median as maximum.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 151, 157, 162, 163, 167, 169

Offset

1,2

Comments

First differs from A334965 in having 750 and lacking 2250.

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

Links

Table of n, a(n) for n=1..62.

Example

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 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 not in the sequence.
The terms together with their prime indices begin:
     1: {}           25: {3,3}           64: {1,1,1,1,1,1}
     2: {1}          27: {2,2,2}         67: {19}
     3: {2}          29: {10}            71: {20}
     4: {1,1}        31: {11}            73: {21}
     5: {3}          32: {1,1,1,1,1}     75: {2,3,3}
     7: {4}          37: {12}            79: {22}
     8: {1,1,1}      41: {13}            81: {2,2,2,2}
     9: {2,2}        43: {14}            83: {23}
    11: {5}          47: {15}            89: {24}
    13: {6}          49: {4,4}           97: {25}
    16: {1,1,1,1}    50: {1,3,3}         98: {1,4,4}
    17: {7}          53: {16}           101: {26}
    18: {1,2,2}      54: {1,2,2,2}      103: {27}
    19: {8}          59: {17}           107: {28}
    23: {9}          61: {18}           108: {1,1,2,2,2}

Mathematica

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

Crossrefs

Partitions of this type are counted by A053263.

For mode instead of median we have A362619, counted by A171979.

For parts at middle position (instead of median) we have A362622.

The complement is A362980, counted by A237821.

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, A237824, A327473, A327476, A359908, A362616, A362620.

Sequence in context: A329131 A362616 A334965 * A133811 A316525 A119314

Adjacent sequences: A362618 A362619 A362620 * A362622 A362623 A362624

Keyword

nonn

Author

Gus Wiseman, May 12 2023

Status

approved