cp's OEIS Frontend

A370594 Number of integer partitions of n such that only one set can be obtained by choosing a different prime factor of each part.

%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