cp's OEIS Frontend

A343344 Number of integer partitions of n that are either empty, or do not have smallest part dividing all the others, but do have greatest part divisible by all the others.

%I A343344 #6 Apr 15 2021 21:43:02
%S A343344 1,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,1,5,1,6,4,6,7,15,6,16,15,20,17,36,18,
%T A343344 43,36,46,48,72,45,93,82,103,88,152,104,179,158,191,194,285,202,328,
%U A343344 292,373,348,502,391,576,519,659,634,864,665
%N A343344 Number of integer partitions of n that are either empty, or do not have smallest part dividing all the others, but do have greatest part divisible by all the others.
%C A343344 Alternative name: Number of integer partitions of n with no part dividing all the others, but with a part divisible by all the others.
%e A343344 The a(18) = 1  through a(23) = 15 partitions (A..E = 10..14):
%e A343344   633222   C43       C332      C432       C64        E72
%e A343344            A522      66332     A5222      A552       F53
%e A343344            C322      633332    C3222      C433       I32
%e A343344            66322     6332222   663222     C3322      C443
%e A343344            633322              6333222    663322     C632
%e A343344            6322222             63222222   6333322    66632
%e A343344                                           63322222   C3332
%e A343344                                                      C4322
%e A343344                                                      663332
%e A343344                                                      A52222
%e A343344                                                      C32222
%e A343344                                                      6333332
%e A343344                                                      6632222
%e A343344                                                      63332222
%e A343344                                                      632222222
%t A343344 Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y A343344 The second condition alone gives A130689.
%Y A343344 The half-opposite versions are A130714 and A343342.
%Y A343344 The first condition alone gives A338470.
%Y A343344 The Heinz numbers of these partitions are 1 and A343339.
%Y A343344 The opposite version is A343345.
%Y A343344 The strict case is A343380.
%Y A343344 A000009 counts strict partitions.
%Y A343344 A000041 counts partitions.
%Y A343344 A000070 counts partitions with a selected part.
%Y A343344 A006128 counts partitions with a selected position.
%Y A343344 A015723 counts strict partitions with a selected part.
%Y A343344 Cf. A083710, A097986, A264401, A339562, A341450, A342193, A343346.
%K A343344 nonn
%O A343344 0,18
%A A343344 _Gus Wiseman_, Apr 15 2021