%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