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Number of strict integer partitions of n that are empty or such that (1) the smallest part divides every other part and (2) the greatest part is divisible by every other part.
12

%I #6 Apr 16 2021 15:45:50

%S 1,1,1,2,2,2,3,3,3,4,4,3,6,5,4,6,6,4,8,6,7,9,8,5,12,9,8,9,11,6,14,10,

%T 10,11,10,10,20,12,12,15,18,10,21,13,15,19,17,11,27,19,20,20,25,13,27,

%U 22,26,23,24,15,34,23,21,27,30,19,38,24,26,27,37

%N Number of strict integer partitions of n that are empty or such that (1) the smallest part divides every other part and (2) the greatest part is divisible by every other part.

%C Alternative name: Number of strict integer partitions of n with a part dividing all the others and a part divisible by all the others.

%e The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):

%e 1 2 3 4 5 6 7 8 9 A B C D E F

%e 21 31 41 42 61 62 63 82 A1 84 C1 C2 A5

%e 51 421 71 81 91 821 93 841 D1 C3

%e 621 631 A2 931 842 E1

%e B1 A21 C21

%e 6321 8421

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

%Y The first condition alone gives A097986.

%Y The non-strict version is A130714 (Heinz numbers are complement of A343343).

%Y The second condition alone gives A343347.

%Y The opposite version is A343379.

%Y The half-opposite versions are A343380 and A343381.

%Y The strict complement is counted by A343382.

%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, A130689, A264401, A339562, A339563, A341450, A343346, A343377.

%K nonn

%O 0,4

%A _Gus Wiseman_, Apr 16 2021