cp's OEIS Frontend

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

%I A382200 #14 Apr 21 2025 17:00:45
%S A382200 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,
%T A382200 34,35,36,37,38,39,41,42,43,44,45,46,47,50,51,52,53,55,57,58,59,60,61,
%U A382200 62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,84
%N A382200 Numbers that can be written as a product of distinct squarefree numbers.
%C A382200 First differs from A339741 in having 1080.
%C A382200 First differs from A382075 in having 18000.
%C A382200 These are positions of positive terms in A050326, complement A293243.
%C A382200 Also numbers whose prime indices can be partitioned into distinct sets.
%C A382200 Differs from A212167, which does not include 18000 = 2^4*3^2*5^3, for example. - _R. J. Mathar_, Mar 23 2025
%H A382200 Robert Israel, <a href="/A382200/b382200.txt">Table of n, a(n) for n = 1..10000</a>
%e A382200 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.
%e A382200 We have 18000 = 2*5*6*10*30, so 18000 is in the sequence.
%p A382200 N:= 1000: # to get all terms <= N
%p A382200 A:= Vector(N):
%p A382200 A[1]:= 1:
%p A382200 for n from 2 to N do
%p A382200   if numtheory:-issqrfree(n) then
%p A382200       S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
%p A382200     fi;
%p A382200 od:
%p A382200 remove(t -> A[t]=0, [$1..N]); # _Robert Israel_, Apr 21 2025
%t A382200 sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t A382200 Select[Range[100],Length[sqfacs[#]]>0&]
%Y A382200 Twice-partitions of this type are counted by A279785, see also A358914.
%Y A382200 Normal multisets not of this type are counted by A292432, strong A292444.
%Y A382200 The complement is A293243, counted by A050342.
%Y A382200 The case of a unique choice is A293511.
%Y A382200 MM-numbers of multiset partitions into distinct sets are A302494.
%Y A382200 For distinct block-sums instead of blocks we have A382075, counted by A381992.
%Y A382200 Partitions of this type are counted by A382077, complement A382078.
%Y A382200 Normal multisets of this type are counted by A382214, strong A381996.
%Y A382200 A001055 counts multiset partitions of prime indices, strict A045778.
%Y A382200 A050320 counts multiset partitions of prime indices into sets.
%Y A382200 A050326 counts multiset partitions of prime indices into distinct sets.
%Y A382200 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A382200 Cf. A000720, A001222, A005117, A050345, A089259, A116539, A270995, A381441, A382201, A382216.
%K A382200 nonn
%O A382200 1,2
%A A382200 _Gus Wiseman_, Mar 21 2025