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Number of strict integer partitions of n with either (1) no part dividing all the others or (2) no part divisible by all the others.
17

%I #5 Apr 16 2021 15:46:20

%S 1,0,0,0,0,1,1,2,3,4,6,9,9,13,18,21,26,34,38,48,57,67,81,99,110,133,

%T 157,183,211,250,282,330,380,437,502,575,648,748,852,967,1095,1250,

%U 1405,1597,1801,2029,2287,2579,2883,3245,3638,4077,4557,5107,5691,6356

%N Number of strict integer partitions of n with either (1) no part dividing all the others or (2) no part divisible by all the others.

%C Alternative name: Number of strict integer partitions of n that are either (1) empty, or (2) have smallest part not dividing all the others, or (3) have greatest part not divisible by all the others.

%e The a(0) = 1 through a(11) = 9 partitions (empty columns indicated by dots):

%e () . . . . (3,2) (3,2,1) (4,3) (5,3) (5,4) (6,4) (6,5)

%e (5,2) (4,3,1) (7,2) (7,3) (7,4)

%e (5,2,1) (4,3,2) (5,3,2) (8,3)

%e (5,3,1) (5,4,1) (9,2)

%e (7,2,1) (5,4,2)

%e (4,3,2,1) (6,3,2)

%e (6,4,1)

%e (7,3,1)

%e (5,3,2,1)

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

%Y The first condition alone gives A341450.

%Y The non-strict version is A343346 (Heinz numbers: A343343).

%Y The second condition alone gives A343377.

%Y The strict complement is A343378.

%Y The version for "and" instead of "or" is A343379.

%Y A000005 counts divisors.

%Y A000009 counts strict 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 A167865 counts strict chains of divisors > 1 summing to n.

%Y A339564 counts factorizations with a selected factor.

%Y Cf. A083710, A097986, A130689, A200745, A264401, A338470, A339562, A342193, A343337, A343338, A343341, A343342.

%K nonn

%O 0,8

%A _Gus Wiseman_, Apr 16 2021