cp's OEIS Frontend

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

%I A293243 #25 Apr 29 2025 23:33:07
%S A293243 4,8,9,16,24,25,27,32,40,48,49,54,56,64,72,80,81,88,96,104,108,112,
%T A293243 121,125,128,135,136,144,152,160,162,169,176,184,189,192,200,208,216,
%U A293243 224,232,240,243,248,250,256,272,288,289,296,297,304,320,324,328,336
%N A293243 Numbers that cannot be written as a product of distinct squarefree numbers.
%C A293243 First differs from A212164 at a(441).
%C A293243 Numbers n such that A050326(n) = 0. - _Felix Fröhlich_, Oct 04 2017
%C A293243 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
%C A293243 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
%H A293243 Robert Israel, <a href="/A293243/b293243.txt">Table of n, a(n) for n = 1..10000</a>
%e A293243 120 is not in the sequence because 120 = 2*6*10. 3600 is not in the sequence because 3600 = 2*6*10*30.
%p A293243 N:= 1000: # to get all terms <= N
%p A293243 A:= Vector(N):
%p A293243 A[1]:= 1:
%p A293243 for n from 2 to N do
%p A293243   if numtheory:-issqrfree(n) then
%p A293243       S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
%p A293243     fi;
%p A293243 od:
%p A293243 select(t -> A[t]=0, [$1..N]); # _Robert Israel_, Oct 10 2017
%t A293243 nn=500;
%t A293243 sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t A293243 Select[Range[nn],Length[sqfacs[#]]===0&]
%Y A293243 These are the zeros of A050326.
%Y A293243 Multiset partitions of this type (set of sets) are counted by A050342.
%Y A293243 Twice-partitions of this type (set of sets) are counted by A279785, see also A358914.
%Y A293243 Normal multisets of this type are counted by A292432, A292444, A381996, A382214.
%Y A293243 The case of a unique choice is A293511, counted by A382079.
%Y A293243 For distinct block-sums instead of blocks see A381806, A381990, A381992, A382075.
%Y A293243 Partitions of this type are counted by A382078.
%Y A293243 The complement is A382200, counted by A382077.
%Y A293243 A001055 counts factorizations, strict A045778.
%Y A293243 A050320 counts factorizations into squarefree numbers.
%Y A293243 A050345 counts factorizations partitioned into into distinct sets.
%Y A293243 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A293243 Cf. A000720, A001222, A005117, A089259, A116539, A246547, A270995, A381441, A382201, A382216.
%K A293243 nonn
%O A293243 1,1
%A A293243 _Gus Wiseman_, Oct 03 2017