A364160 Numbers whose least prime factor has the greatest exponent.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 52, 53, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 76, 79, 80, 81, 83, 84, 88, 89, 92, 96, 97, 99, 101, 103, 104, 107, 109, 112, 113, 116

Offset: 1

Author

Gus Wiseman, Jul 14 2023

Keywords

• nonn

Comments

First differs from A334298 in having 600 and lacking 180.

Also numbers whose minimum part in prime factorization is a unique mode.

If k is a term, then so are all powers of k. - Robert Israel, Sep 17 2024

Examples

			The prime factorization of 600 is 2*2*2*3*5*5, and 3 > max(1,2), so 600 is in the sequence.
The prime factorization of 180 is 2*2*3*3*5, but 2 <= max(2,1), so 180 is not in the sequence.
The terms together with their prime indices begin:
     1: {}           29: {10}              67: {19}
     2: {1}          31: {11}              68: {1,1,7}
     3: {2}          32: {1,1,1,1,1}       71: {20}
     4: {1,1}        37: {12}              72: {1,1,1,2,2}
     5: {3}          40: {1,1,1,3}         73: {21}
     7: {4}          41: {13}              76: {1,1,8}
     8: {1,1,1}      43: {14}              79: {22}
     9: {2,2}        44: {1,1,5}           80: {1,1,1,1,3}
    11: {5}          45: {2,2,3}           81: {2,2,2,2}
    12: {1,1,2}      47: {15}              83: {23}
    13: {6}          48: {1,1,1,1,2}       84: {1,1,2,4}
    16: {1,1,1,1}    49: {4,4}             88: {1,1,1,5}
    17: {7}          52: {1,1,6}           89: {24}
    19: {8}          53: {16}              92: {1,1,9}
    20: {1,1,3}      56: {1,1,1,4}         96: {1,1,1,1,1,2}
    23: {9}          59: {17}              97: {25}
    24: {1,1,1,2}    60: {1,1,2,3}         99: {2,2,5}
    25: {3,3}        61: {18}             101: {26}
    27: {2,2,2}      63: {2,2,4}          103: {27}
    28: {1,1,4}      64: {1,1,1,1,1,1}    104: {1,1,1,6}
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Allowing any unique mode gives A356862, complement A362605.

Allowing any unique co-mode gives A359178, complement A362606.

The even case is A360013, counted by A241131.

For greatest instead of least we have A362616, counted by A362612.

These partitions are counted by A364193.

A027746 lists prime factors (with multiplicity).

A112798 lists prime indices, length A001222, sum A056239.

A362611 counts modes in prime factorization, triangle A362614.

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

A363486 gives least mode in prime indices, A363487 greatest.

Cf. A098859, A327473, A327476, A360014, A360015, A362610, A364061, A364062.

Programs

Maple

filter:= proc(n) local F,i;
  F:= ifactors(n)[2];
  if nops(F) = 1 then return true fi;
  i:= min[index](F[..,1]);
  andmap(t -> F[t,2] < F[i,2], {$1..nops(F)} minus {i})
end proc:
filter(1):= true:
select(filter, [$1..200]); # Robert Israel, Sep 17 2024

Mathematica

Select[Range[100],First[Last/@FactorInteger[#]] > Max@@Rest[Last/@FactorInteger[#]]&]