cp's OEIS Frontend

A343382 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.

%I A343382 #5 Apr 16 2021 15:46:20
%S A343382 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 A343382 157,183,211,250,282,330,380,437,502,575,648,748,852,967,1095,1250,
%U A343382 1405,1597,1801,2029,2287,2579,2883,3245,3638,4077,4557,5107,5691,6356
%N A343382 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 A343382 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 A343382 The a(0) = 1 through a(11) = 9 partitions (empty columns indicated by dots):
%e A343382   ()  .  .  .  .  (3,2)  (3,2,1)  (4,3)  (5,3)    (5,4)    (6,4)      (6,5)
%e A343382                                   (5,2)  (4,3,1)  (7,2)    (7,3)      (7,4)
%e A343382                                          (5,2,1)  (4,3,2)  (5,3,2)    (8,3)
%e A343382                                                   (5,3,1)  (5,4,1)    (9,2)
%e A343382                                                            (7,2,1)    (5,4,2)
%e A343382                                                            (4,3,2,1)  (6,3,2)
%e A343382                                                                       (6,4,1)
%e A343382                                                                       (7,3,1)
%e A343382                                                                       (5,3,2,1)
%t A343382 Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)||UnsameQ@@#&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y A343382 The first condition alone gives A341450.
%Y A343382 The non-strict version is A343346 (Heinz numbers: A343343).
%Y A343382 The second condition alone gives A343377.
%Y A343382 The strict complement is A343378.
%Y A343382 The version for "and" instead of "or" is A343379.
%Y A343382 A000005 counts divisors.
%Y A343382 A000009 counts strict partitions.
%Y A343382 A000070 counts partitions with a selected part.
%Y A343382 A006128 counts partitions with a selected position.
%Y A343382 A015723 counts strict partitions with a selected part.
%Y A343382 A018818 counts partitions into divisors (strict: A033630).
%Y A343382 A167865 counts strict chains of divisors > 1 summing to n.
%Y A343382 A339564 counts factorizations with a selected factor.
%Y A343382 Cf. A083710, A097986, A130689, A200745, A264401, A338470, A339562, A342193, A343337, A343338, A343341, A343342.
%K A343382 nonn
%O A343382 0,8
%A A343382 _Gus Wiseman_, Apr 16 2021