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%I #6 Apr 15 2021 21:43:18
%S 1,0,0,0,0,1,1,4,6,11,16,29,36,59,80,112,150,214,271,374,476,624,800,
%T 1045,1298,1669,2088,2628,3258,4087,5000,6219,7602,9331,11368,13877,
%U 16754,20368,24536,29580,35468,42624,50845,60827,72357,86078,102100,121101
%N Number of integer partitions of n that are empty, have smallest part not dividing all the others, or greatest part not divisible by all the others.
%C First differs from A343345 at a(14) = 80, A343345(14) = 79.
%C Alternative name: Number of integer partitions of n with either no part dividing, or no part divisible by all the others.
%e The a(0) = 1 through a(10) = 16 partitions (empty columns indicated by dots):
%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/@(#/Min@@#)||!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y The complement is counted by A130714.
%Y The first condition alone gives A338470.
%Y The second condition alone gives A343341.
%Y The "and" instead of "or" version is A343342.
%Y The Heinz numbers of these partitions are A343343.
%Y The strict case is A343382.
%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. A083710, A130689, A341450, A342193, A343337, A343338, A343379.
%K nonn
%O 0,8
%A _Gus Wiseman_, Apr 15 2021
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