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%I #6 Apr 15 2021 21:43:02
%S 1,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,1,5,1,6,4,6,7,15,6,16,15,20,17,36,18,
%T 43,36,46,48,72,45,93,82,103,88,152,104,179,158,191,194,285,202,328,
%U 292,373,348,502,391,576,519,659,634,864,665
%N Number of integer partitions of n that are either empty, or do not have smallest part dividing all the others, but do have greatest part divisible by all the others.
%C Alternative name: Number of integer partitions of n with no part dividing all the others, but with a part divisible by all the others.
%e The a(18) = 1 through a(23) = 15 partitions (A..E = 10..14):
%e 633222 C43 C332 C432 C64 E72
%e A522 66332 A5222 A552 F53
%e C322 633332 C3222 C433 I32
%e 66322 6332222 663222 C3322 C443
%e 633322 6333222 663322 C632
%e 6322222 63222222 6333322 66632
%e 63322222 C3332
%e C4322
%e 663332
%e A52222
%e C32222
%e 6333332
%e 6632222
%e 63332222
%e 632222222
%t Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y The second condition alone gives A130689.
%Y The half-opposite versions are A130714 and A343342.
%Y The first condition alone gives A338470.
%Y The Heinz numbers of these partitions are 1 and A343339.
%Y The opposite version is A343345.
%Y The strict case is A343380.
%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, A097986, A264401, A339562, A341450, A342193, A343346.
%K nonn
%O 0,18
%A _Gus Wiseman_, Apr 15 2021
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