A362616 Numbers in whose prime factorization the greatest factor is the unique mode.

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, 150, 151, 157, 162, 163, 167

Offset

1,1

Comments

First differs from A329131 in lacking 450 and having 1500.

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

Links

Table of n, a(n) for n=1..61.

Example

The factorization of 90 is 2*3*3*5, modes {3}, so 90 is missing.
The factorization of 450 is 2*3*3*5*5, modes {3,5}, so 450 is missing.
The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so 900 is missing.
The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so 1500 is present.
The terms together with their prime indices begin:
     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}
    25: {3,3}        64: {1,1,1,1,1,1}    109: {29}

Mathematica

prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100], Commonest[prifacs[#]]=={Max[prifacs[#]]}&]

Crossrefs

First term with given bigomega is A000079.

For median instead of mode we have A053263.

Partitions of this type are counted by A362612.

A112798 lists prime indices, length A001222, sum A056239.

A356862 ranks partitions with a unique mode, counted by A362608.

A359178 ranks partitions with a unique co-mode, counted by A362610.

A362605 ranks partitions with more than one mode, counted by A362607.

A362606 ranks partitions with more than one co-mode, counted by A362609.

A362614 counts partitions by number of modes, ranked by A362611.

A362615 counts partitions by number of co-modes, ranked by A362613.

Cf. A000040, A002865, A327473, A327476, A358137, A360687.

Sequence in context: A082377 A166401 A329131 * A334965 A362621 A133811

Adjacent sequences: A362613 A362614 A362615 * A362617 A362618 A362619

Keyword

nonn

Author

Gus Wiseman, May 05 2023

Status

approved