cp's OEIS Frontend

A343345 Number of integer partitions of n that are empty, or have smallest part dividing all the others, but do not have greatest part divisible by all the others.

%I A343345 #12 Apr 17 2021 01:57:24
%S A343345 1,0,0,0,0,0,1,1,4,6,11,16,29,36,59,79,115,149,216,270,379,473,634,
%T A343345 793,1063,1292,1689,2079,2667,3241,4142,4982,6291,7582,9434,11321,
%U A343345 14049,16709,20545,24490,29860,35380,43004,50741,61282,72284,86680,101906,121990
%N A343345 Number of integer partitions of n that are empty, or have smallest part dividing all the others, but do not have greatest part divisible by all the others.
%C A343345 First differs from A343346 at a(14) = 79, A343346(14) = 80.
%C A343345 Alternative name: Number of integer partitions of n with a part dividing all the others, but with no part divisible by all the others.
%e A343345 The a(6) = 1 through a(11) = 16 partitions:
%e A343345   (321)  (3211)  (431)    (531)     (541)      (641)
%e A343345                  (521)    (3321)    (721)      (731)
%e A343345                  (3221)   (4311)    (4321)     (4331)
%e A343345                  (32111)  (5211)    (5221)     (5321)
%e A343345                           (32211)   (5311)     (5411)
%e A343345                           (321111)  (32221)    (7211)
%e A343345                                     (33211)    (33221)
%e A343345                                     (43111)    (43211)
%e A343345                                     (52111)    (52211)
%e A343345                                     (322111)   (53111)
%e A343345                                     (3211111)  (322211)
%e A343345                                                (332111)
%e A343345                                                (431111)
%e A343345                                                (521111)
%e A343345                                                (3221111)
%e A343345                                                (32111111)
%t A343345 Table[Length[Select[IntegerPartitions[n],#=={}||And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y A343345 The first condition alone gives A083710.
%Y A343345 The half-opposite versions are A130714 and A343342.
%Y A343345 The Heinz numbers of these partitions are 1 and A343340.
%Y A343345 The second condition alone gives A343341.
%Y A343345 The opposite version is A343344.
%Y A343345 The strict case is A343381.
%Y A343345 A000009 counts strict partitions.
%Y A343345 A000041 counts partitions.
%Y A343345 A000070 counts partitions with a selected part.
%Y A343345 A006128 counts partitions with a selected position.
%Y A343345 A015723 counts strict partitions with a selected part.
%Y A343345 A018818 counts partitions into divisors (strict: A033630).
%Y A343345 Cf. A083711, A097986, A098743, A098965, A130689, A264401, A339563, A343337.
%K A343345 nonn
%O A343345 0,9
%A A343345 _Gus Wiseman_, Apr 15 2021