cp's OEIS Frontend

A343346 Number of integer partitions of n that are empty, have smallest part not dividing all the others, or greatest part not divisible by all the others.

%I A343346 #6 Apr 15 2021 21:43:18
%S A343346 1,0,0,0,0,1,1,4,6,11,16,29,36,59,80,112,150,214,271,374,476,624,800,
%T A343346 1045,1298,1669,2088,2628,3258,4087,5000,6219,7602,9331,11368,13877,
%U A343346 16754,20368,24536,29580,35468,42624,50845,60827,72357,86078,102100,121101
%N A343346 Number of integer partitions of n that are empty, have smallest part not dividing all the others, or greatest part not divisible by all the others.
%C A343346 First differs from A343345 at a(14) = 80, A343345(14) = 79.
%C A343346 Alternative name: Number of integer partitions of n with either no part dividing, or no part divisible by all the others.
%e A343346 The a(0) = 1 through a(10) = 16 partitions (empty columns indicated by dots):
%e A343346   ()  .  .  .  .  (32)  (321)  (43)    (53)     (54)      (64)
%e A343346                                (52)    (332)    (72)      (73)
%e A343346                                (322)   (431)    (432)     (433)
%e A343346                                (3211)  (521)    (522)     (532)
%e A343346                                        (3221)   (531)     (541)
%e A343346                                        (32111)  (3222)    (721)
%e A343346                                                 (3321)    (3322)
%e A343346                                                 (4311)    (4321)
%e A343346                                                 (5211)    (5221)
%e A343346                                                 (32211)   (5311)
%e A343346                                                 (321111)  (32221)
%e A343346                                                           (33211)
%e A343346                                                           (43111)
%e A343346                                                           (52111)
%e A343346                                                           (322111)
%e A343346                                                           (3211111)
%t A343346 Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)||!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y A343346 The complement is counted by A130714.
%Y A343346 The first condition alone gives A338470.
%Y A343346 The second condition alone gives A343341.
%Y A343346 The "and" instead of "or" version is A343342.
%Y A343346 The Heinz numbers of these partitions are A343343.
%Y A343346 The strict case is A343382.
%Y A343346 A000009 counts strict partitions.
%Y A343346 A000041 counts partitions.
%Y A343346 A000070 counts partitions with a selected part.
%Y A343346 A006128 counts partitions with a selected position.
%Y A343346 A015723 counts strict partitions with a selected part.
%Y A343346 Cf. A083710, A130689, A341450, A342193, A343337, A343338, A343379.
%K A343346 nonn
%O A343346 0,8
%A A343346 _Gus Wiseman_, Apr 15 2021