|
%I #6 Apr 16 2021 15:45:57
%S 1,0,0,0,0,1,0,2,1,3,3,5,3,9,9,12,12,18,18,27,27,36,41,51,51,73,80,96,
%T 105,132,137,177,188,230,253,303,320,398,431,508,550,659,705,847,913,
%U 1063,1165,1359,1452,1716,1856,2134,2329,2688,2894,3345,3622,4133
%N Number of strict integer partitions of n with no part dividing or divisible by all the other parts.
%C Alternative name: Number of strict integer partitions of n that are either empty, or (1) have smallest part not dividing all the others and (2) have greatest part not divisible by all the others.
%F The Heinz numbers for the non-strict version are A343338 = A342193 /\ A343337.
%e The a(5) = 1 through a(13) = 9 partitions (empty column indicated by dot):
%e (3,2) . (4,3) (5,3) (5,4) (6,4) (6,5) (7,5) (7,6)
%e (5,2) (7,2) (7,3) (7,4) (5,4,3) (8,5)
%e (4,3,2) (5,3,2) (8,3) (7,3,2) (9,4)
%e (9,2) (10,3)
%e (5,4,2) (11,2)
%e (6,4,3)
%e (6,5,2)
%e (7,4,2)
%e (8,3,2)
%t Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]
%Y The first condition alone gives A341450.
%Y The non-strict version is A343342 (Heinz numbers: A343338).
%Y The second condition alone gives A343377.
%Y The opposite version is A343378.
%Y The half-opposite versions are A343380 and A343381.
%Y The version for "or" instead of "and" is A343382.
%Y A000009 counts strict partitions.
%Y A000070 counts partitions with a selected part.
%Y A006128 counts partitions with a selected position.
%Y A015723 counts strict partitions with a selected part.
%Y A018818 counts partitions into divisors (strict: A033630).
%Y A167865 counts strict chains of divisors > 1 summing to n.
%Y A339564 counts factorizations with a selected factor.
%Y Cf. A083710, A097986, A200745, A264401, A338470, A339562, A342193, A343337, A343341, A343343, A343346, A343347.
%K nonn
%O 0,8
%A _Gus Wiseman_, Apr 16 2021
|
