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Number of integer partitions of n with no part dividing or divisible by all the others.
14

%I #6 Apr 15 2021 21:42:54

%S 1,0,0,0,0,1,0,3,2,5,5,12,7,22,20,32,34,60,54,98,93,145,159,237,229,

%T 361,384,529,574,810,840,1194,1275,1703,1886,2484,2660,3566,3909,4987,

%U 5520,7092,7737,9907,10917,13603,15226,18910,20801,25912,28797

%N Number of integer partitions of n with no part dividing or divisible by all the others.

%C Alternative name: Number of integer partitions of n that are either empty or have smallest part not dividing all the others and greatest part not divisible by all the others.

%e The a(0) = 1 through a(12) = 7 partitions (empty columns indicated by dots):

%e () . . . . (32) . (43) (53) (54) (64) (65) (75)

%e (52) (332) (72) (73) (74) (543)

%e (322) (432) (433) (83) (552)

%e (522) (532) (92) (732)

%e (3222) (3322) (443) (4332)

%e (533) (5322)

%e (542) (33222)

%e (722)

%e (3332)

%e (4322)

%e (5222)

%e (32222)

%t Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

%Y The opposite version is A130714.

%Y The first condition alone gives A338470.

%Y The Heinz numbers of these partitions are A343338 = A342193 /\ A343337.

%Y The second condition alone gives A343341.

%Y The half-opposite versions are A343344 and A343345.

%Y The "or" instead of "and" version is A343346 (strict: A343382).

%Y The strict case is A343379.

%Y A000009 counts strict partitions.

%Y A000041 counts partitions.

%Y A000070 counts partitions with a selected part (strict: A015723).

%Y Cf. A006128, A066186, A083710, A130689, A341450, A343343, A343377.

%K nonn

%O 0,8

%A _Gus Wiseman_, Apr 15 2021