%I #12 Apr 17 2021 01:57:24
%S 1,0,0,0,0,0,1,1,4,6,11,16,29,36,59,79,115,149,216,270,379,473,634,
%T 793,1063,1292,1689,2079,2667,3241,4142,4982,6291,7582,9434,11321,
%U 14049,16709,20545,24490,29860,35380,43004,50741,61282,72284,86680,101906,121990
%N Number of integer partitions of n that are empty, or have smallest part dividing all the others, but do not have greatest part divisible by all the others.
%C First differs from A343346 at a(14) = 79, A343346(14) = 80.
%C Alternative name: Number of integer partitions of n with a part dividing all the others, but with no part divisible by all the others.
%e The a(6) = 1 through a(11) = 16 partitions:
%e (321) (3211) (431) (531) (541) (641)
%e (521) (3321) (721) (731)
%e (3221) (4311) (4321) (4331)
%e (32111) (5211) (5221) (5321)
%e (32211) (5311) (5411)
%e (321111) (32221) (7211)
%e (33211) (33221)
%e (43111) (43211)
%e (52111) (52211)
%e (322111) (53111)
%e (3211111) (322211)
%e (332111)
%e (431111)
%e (521111)
%e (3221111)
%e (32111111)
%t Table[Length[Select[IntegerPartitions[n],#=={}||And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y The first condition alone gives A083710.
%Y The half-opposite versions are A130714 and A343342.
%Y The Heinz numbers of these partitions are 1 and A343340.
%Y The second condition alone gives A343341.
%Y The opposite version is A343344.
%Y The strict case is A343381.
%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 A018818 counts partitions into divisors (strict: A033630).
%Y Cf. A083711, A097986, A098743, A098965, A130689, A264401, A339563, A343337.
%K nonn
%O 0,9
%A _Gus Wiseman_, Apr 15 2021