A293243 Numbers that cannot be written as a product of distinct squarefree numbers.

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Author

Gus Wiseman, Oct 03 2017

Keywords

• nonn

Comments

First differs from A212164 at a(441).

Numbers n such that A050326(n) = 0. - Felix Fröhlich, Oct 04 2017

Includes A246547, and all numbers of the form p^a*q^b where p and q are primes, a >= 1 and b >= 3. - Robert Israel, Oct 10 2017

Also numbers whose prime indices cannot be partitioned into a set of sets. For example, the prime indices of 90 are {1,2,2,3}, and we have sets of sets: {{2},{1,2,3}}, {{1,2},{2,3}}, {{1},{2},{2,3}}, {{2},{3},{1,2}}, so 90 is not in the sequence. - Gus Wiseman, Apr 28 2025

Examples

			120 is not in the sequence because 120 = 2*6*10. 3600 is not in the sequence because 3600 = 2*6*10*30.
		

Links

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

Crossrefs

These are the zeros of A050326.

Multiset partitions of this type (set of sets) are counted by A050342.

Twice-partitions of this type (set of sets) are counted by A279785, see also A358914.

Normal multisets of this type are counted by A292432, A292444, A381996, A382214.

The case of a unique choice is A293511, counted by A382079.

For distinct block-sums instead of blocks see A381806, A381990, A381992, A382075.

Partitions of this type are counted by A382078.

The complement is A382200, counted by A382077.

A001055 counts factorizations, strict A045778.

A050320 counts factorizations into squarefree numbers.

A050345 counts factorizations partitioned into into distinct sets.

A317141 counts coarsenings of prime indices, refinements A300383.

Cf. A000720, A001222, A005117, A089259, A116539, A246547, 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:
select(t -> A[t]=0, [$1..N]); # Robert Israel, Oct 10 2017

Mathematica

nn=500;
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[nn],Length[sqfacs[#]]===0&]