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%I #5 Apr 15 2021 21:42:45
%S 1,0,0,0,0,1,1,4,6,11,16,28,36,58,79,111,149,209,270,368,472,618,793,
%T 1030,1292,1653,2073,2608,3241,4051,4982,6176,7566,9285,11320,13805,
%U 16709,20275,24454,29477,35380,42472,50741,60648,72199,85887,101906,120816
%N Number of integer partitions of n with no part divisible by all the others.
%C Alternative name: Number of integer partitions of n that are either empty, or have greatest part not divisible by all the others.
%e The a(5) = 1 through a(10) = 16 partitions:
%e (32) (321) (43) (53) (54) (64)
%e (52) (332) (72) (73)
%e (322) (431) (432) (433)
%e (3211) (521) (522) (532)
%e (3221) (531) (541)
%e (32111) (3222) (721)
%e (3321) (3322)
%e (4311) (4321)
%e (5211) (5221)
%e (32211) (5311)
%e (321111) (32221)
%e (33211)
%e (43111)
%e (52111)
%e (322111)
%e (3211111)
%t Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y The complement is counted by A130689.
%Y The dual version is A338470.
%Y The Heinz numbers of these partitions are A343337.
%Y The strict case is A343377.
%Y A000009 counts strict partitions.
%Y A000041 counts partitions.
%Y A000070 counts partitions with a selected part.
%Y A006128 counts partitions with a selected position.
%Y A015723 counts strict partitions with a selected part.
%Y Cf. A066186, A083710, A083711, A097986, A098965, A341450, A343342, A343345, A343346, A343381, A343382.
%K nonn
%O 0,8
%A _Gus Wiseman_, Apr 15 2021
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