# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a370594 Showing 1-1 of 1 %I A370594 #7 Mar 01 2024 09:34:58 %S A370594 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 A370594 49,55,53,72,80,90,92,110,110,132,154,169,180,201,218,246,281,302,323, %U A370594 348,396,433,482,530,584,618,670,754,823,903,980,1047,1137 %N A370594 Number of integer partitions of n such that only one set can be obtained by choosing a different prime factor of each part. %e A370594 The partition (10,6,4) has unique choice (5,3,2) so is counted under a(20). %e A370594 The a(0) = 1 through a(12) = 5 partitions: %e A370594 () . (2) (3) (4) (5) . (7) (8) (9) (6,4) (11) (6,6) %e A370594 (3,2) (4,3) (5,3) (5,4) (7,3) (7,4) (7,5) %e A370594 (5,2) (6,2) (6,3) (5,3,2) (8,3) (10,2) %e A370594 (7,2) (9,2) (5,4,3) %e A370594 (7,3,2) %t A370594 Table[Length[Select[IntegerPartitions[n], Length[Union[Sort/@Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]],{n,0,30}] %Y A370594 The version for set-systems is A367904, ranks A367908. %Y A370594 Multisets of this type are ranked by A368101, cf. A368100, A355529. %Y A370594 The version for subsets is A370584, cf. A370582, A370583, A370586, A370587. %Y A370594 Maximal sets of this type are counted by A370585. %Y A370594 For existence we have A370592. %Y A370594 For nonexistence we have A370593. %Y A370594 For divisors instead of factors we have A370595. %Y A370594 For subsets and binary indices we have A370638, cf. A370636, A370637. %Y A370594 The version for factorizations is A370645, cf. A368414, A368413. %Y A370594 These partitions have ranks A370647. %Y A370594 A006530 gives greatest prime factor, least A020639. %Y A370594 A027746 lists prime factors, A112798 indices, length A001222. %Y A370594 A355741 counts ways to choose a prime factor of each prime index. %Y A370594 Cf. A000040, A000041, A000720, A003963, A355739, A355740, A367905, A368110. %K A370594 nonn %O A370594 0,6 %A A370594 _Gus Wiseman_, Feb 29 2024 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE