A382200 Numbers that can be written as a product of distinct squarefree numbers.

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84

Offset: 1

Author

Gus Wiseman, Mar 21 2025

Keywords

• nonn

Comments

First differs from A339741 in having 1080.

First differs from A382075 in having 18000.

These are positions of positive terms in A050326, complement A293243.

Also numbers whose prime indices can be partitioned into distinct sets.

Differs from A212167, which does not include 18000 = 2^4*3^2*5^3, for example. - R. J. Mathar, Mar 23 2025

Examples

			The prime indices of 1080 are {1,1,1,2,2,2,3}, and {{1},{2},{1,2},{1,2,3}} is a partition into a set of sets, so 1080 is in the sequence.
We have 18000 = 2*5*6*10*30, so 18000 is in the sequence.
		

Links

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

Crossrefs

Twice-partitions of this type are counted by A279785, see also A358914.

Normal multisets not of this type are counted by A292432, strong A292444.

The complement is A293243, counted by A050342.

The case of a unique choice is A293511.

MM-numbers of multiset partitions into distinct sets are A302494.

For distinct block-sums instead of blocks we have A382075, counted by A381992.

Partitions of this type are counted by A382077, complement A382078.

Normal multisets of this type are counted by A382214, strong A381996.

A001055 counts multiset partitions of prime indices, strict A045778.

A050320 counts multiset partitions of prime indices into sets.

A050326 counts multiset partitions of prime indices into distinct sets.

A317141 counts coarsenings of prime indices, refinements A300383.

Cf. A000720, A001222, A005117, A050345, A089259, A116539, A270995, A381441, A382201, A382216.

Programs

Maple

N:= 1000: # to get all terms <= N
A:= Vector(N):
A[1]:= 1:
for n from 2 to N do
  if numtheory:-issqrfree(n) then
      S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
    fi;
od:
remove(t -> A[t]=0, [$1..N]); # Robert Israel, Apr 21 2025

Mathematica

sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[100],Length[sqfacs[#]]>0&]