A343380 - OEIS

%I #6 Apr 16 2021 15:46:04

%S 1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,1,0,1,1,4,0,1,0,2,0,4,0,3,1,2,

%T 2,5,0,5,3,4,1,9,1,5,2,4,5,11,1,6,4,11,3,13,5,10,4,11,8,14,3,10,6,9,3,

%U 15,6,14,10,18,8

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

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

%e The a(11) = 1 through a(29) = 4 partitions (empty columns indicated by dots, A..O = 10..24):

%e   632  .  .  .  .  .  A52  .  C43  .  C432  C64  E72   .  C643  .  K52    .  I92

%e                       C32                        F53               C6432     K54

%e                                                  I32                         O32

%e                                                  C632                        I632

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

%Y The first condition alone gives A341450.

%Y The non-strict version is A343344 (Heinz numbers: A343339).

%Y The second condition alone gives A343347.

%Y The half-opposite versions are A343378 and A343379.

%Y The opposite (and dual) version is A343381.

%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 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, A343377, A343382.

%K nonn

%O 0,18

%A _Gus Wiseman_, Apr 16 2021