cp's OEIS Frontend

A343378 Number of strict integer partitions of n that are empty or such that (1) the smallest part divides every other part and (2) the greatest part is divisible by every other part.

%I A343378 #6 Apr 16 2021 15:45:50
%S A343378 1,1,1,2,2,2,3,3,3,4,4,3,6,5,4,6,6,4,8,6,7,9,8,5,12,9,8,9,11,6,14,10,
%T A343378 10,11,10,10,20,12,12,15,18,10,21,13,15,19,17,11,27,19,20,20,25,13,27,
%U A343378 22,26,23,24,15,34,23,21,27,30,19,38,24,26,27,37
%N A343378 Number of strict integer partitions of n that are empty or such that (1) the smallest part divides every other part and (2) the greatest part is divisible by every other part.
%C A343378 Alternative name: Number of strict integer partitions of n with a part dividing all the others and a part divisible by all the others.
%e A343378 The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
%e A343378   1  2  3   4   5   6   7    8   9    A    B    C     D    E    F
%e A343378         21  31  41  42  61   62  63   82   A1   84    C1   C2   A5
%e A343378                     51  421  71  81   91   821  93    841  D1   C3
%e A343378                                  621  631       A2    931  842  E1
%e A343378                                                 B1    A21       C21
%e A343378                                                 6321            8421
%t A343378 Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&And@@IntegerQ/@(#/Min@@#)&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y A343378 The first condition alone gives A097986.
%Y A343378 The non-strict version is A130714 (Heinz numbers are complement of A343343).
%Y A343378 The second condition alone gives A343347.
%Y A343378 The opposite version is A343379.
%Y A343378 The half-opposite versions are A343380 and A343381.
%Y A343378 The strict complement is counted by A343382.
%Y A343378 A000009 counts strict partitions.
%Y A343378 A000070 counts partitions with a selected part.
%Y A343378 A006128 counts partitions with a selected position.
%Y A343378 A015723 counts strict partitions with a selected part.
%Y A343378 A018818 counts partitions into divisors (strict: A033630).
%Y A343378 A167865 counts strict chains of divisors > 1 summing to n.
%Y A343378 A339564 counts factorizations with a selected factor.
%Y A343378 Cf. A083710, A130689, A264401, A339562, A339563, A341450, A343346, A343377.
%K A343378 nonn
%O A343378 0,4
%A A343378 _Gus Wiseman_, Apr 16 2021