%I #7 Mar 01 2024 09:34:58

%S 1,0,1,1,1,2,0,3,3,4,3,4,5,5,8,10,11,7,14,13,19,23,24,20,30,33,40,47,

%T 49,55,53,72,80,90,92,110,110,132,154,169,180,201,218,246,281,302,323,

%U 348,396,433,482,530,584,618,670,754,823,903,980,1047,1137

%N Number of integer partitions of n such that only one set can be obtained by choosing a different prime factor of each part.

%e The partition (10,6,4) has unique choice (5,3,2) so is counted under a(20).

%e The a(0) = 1 through a(12) = 5 partitions:

%e () . (2) (3) (4) (5) . (7) (8) (9) (6,4) (11) (6,6)

%e (3,2) (4,3) (5,3) (5,4) (7,3) (7,4) (7,5)

%e (5,2) (6,2) (6,3) (5,3,2) (8,3) (10,2)

%e (7,2) (9,2) (5,4,3)

%e (7,3,2)

%t Table[Length[Select[IntegerPartitions[n], Length[Union[Sort/@Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]],{n,0,30}]

%Y The version for set-systems is A367904, ranks A367908.

%Y Multisets of this type are ranked by A368101, cf. A368100, A355529.

%Y The version for subsets is A370584, cf. A370582, A370583, A370586, A370587.

%Y Maximal sets of this type are counted by A370585.

%Y For existence we have A370592.

%Y For nonexistence we have A370593.

%Y For divisors instead of factors we have A370595.

%Y For subsets and binary indices we have A370638, cf. A370636, A370637.

%Y The version for factorizations is A370645, cf. A368414, A368413.

%Y These partitions have ranks A370647.

%Y A006530 gives greatest prime factor, least A020639.

%Y A027746 lists prime factors, A112798 indices, length A001222.

%Y A355741 counts ways to choose a prime factor of each prime index.

%Y Cf. A000040, A000041, A000720, A003963, A355739, A355740, A367905, A368110.

%K nonn

%O 0,6

%A _Gus Wiseman_, Feb 29 2024