cp's OEIS Frontend

A343342 Number of integer partitions of n with no part dividing or divisible by all the others.

%I A343342 #6 Apr 15 2021 21:42:54
%S A343342 1,0,0,0,0,1,0,3,2,5,5,12,7,22,20,32,34,60,54,98,93,145,159,237,229,
%T A343342 361,384,529,574,810,840,1194,1275,1703,1886,2484,2660,3566,3909,4987,
%U A343342 5520,7092,7737,9907,10917,13603,15226,18910,20801,25912,28797
%N A343342 Number of integer partitions of n with no part dividing or divisible by all the others.
%C A343342 Alternative name: Number of integer partitions of n that are either empty or have smallest part not dividing all the others and greatest part not divisible by all the others.
%e A343342 The a(0) = 1 through a(12) = 7 partitions (empty columns indicated by dots):
%e A343342   ()  .  .  .  .  (32)  .  (43)   (53)   (54)    (64)    (65)     (75)
%e A343342                            (52)   (332)  (72)    (73)    (74)     (543)
%e A343342                            (322)         (432)   (433)   (83)     (552)
%e A343342                                          (522)   (532)   (92)     (732)
%e A343342                                          (3222)  (3322)  (443)    (4332)
%e A343342                                                          (533)    (5322)
%e A343342                                                          (542)    (33222)
%e A343342                                                          (722)
%e A343342                                                          (3332)
%e A343342                                                          (4322)
%e A343342                                                          (5222)
%e A343342                                                          (32222)
%t A343342 Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y A343342 The opposite version is A130714.
%Y A343342 The first condition alone gives A338470.
%Y A343342 The Heinz numbers of these partitions are A343338 = A342193 /\ A343337.
%Y A343342 The second condition alone gives A343341.
%Y A343342 The half-opposite versions are A343344 and A343345.
%Y A343342 The "or" instead of "and" version is A343346 (strict: A343382).
%Y A343342 The strict case is A343379.
%Y A343342 A000009 counts strict partitions.
%Y A343342 A000041 counts partitions.
%Y A343342 A000070 counts partitions with a selected part (strict: A015723).
%Y A343342 Cf. A006128, A066186, A083710, A130689, A341450, A343343, A343377.
%K A343342 nonn
%O A343342 0,8
%A A343342 _Gus Wiseman_, Apr 15 2021