cp's OEIS Frontend

A370593 Number of integer partitions of n such that it is not possible to choose a different prime factor of each part.

%I A370593 #6 Mar 01 2024 09:35:59
%S A370593 0,1,1,2,4,5,10,12,19,26,38,51,71,94,126,165,219,285,369,472,605,766,
%T A370593 973,1226,1538,1917,2387,2955,3657,4497,5532,6754,8251,10033,12190,
%U A370593 14748,17831,21471,25825,30976,37111,44331,52897,62952,74829,88755,105145,124307
%N A370593 Number of integer partitions of n such that it is not possible to choose a different prime factor of each part.
%F A370593 a(n) = A000041(n) - A370592(n).
%e A370593 The a(0) = 0 through a(7) = 12 partitions:
%e A370593   .  (1)  (11)  (21)   (22)    (41)     (33)      (61)
%e A370593                 (111)  (31)    (221)    (42)      (322)
%e A370593                        (211)   (311)    (51)      (331)
%e A370593                        (1111)  (2111)   (222)     (421)
%e A370593                                (11111)  (321)     (511)
%e A370593                                         (411)     (2221)
%e A370593                                         (2211)    (3211)
%e A370593                                         (3111)    (4111)
%e A370593                                         (21111)   (22111)
%e A370593                                         (111111)  (31111)
%e A370593                                                   (211111)
%e A370593                                                   (1111111)
%t A370593 Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]==0&]],{n,0,30}]
%Y A370593 The complement for divisors instead of factors is A239312, ranks A368110.
%Y A370593 These partitions have ranks A355529, complement A368100.
%Y A370593 The complement for set-systems is A367902, ranks A367906, unlabeled A368095.
%Y A370593 The version for set-systems is A367903, ranks A367907, unlabeled A368094.
%Y A370593 For unlabeled multiset partitions we have A368097, complement A368098.
%Y A370593 The version for factorizations is A368413, complement A368414.
%Y A370593 The complement is counted by A370592.
%Y A370593 For a unique choice we have A370594, ranks A370647.
%Y A370593 A006530 gives greatest prime factor, least A020639.
%Y A370593 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370593 A355741 counts choices of a prime factor of each prime index.
%Y A370593 Cf. A000040, A000720, A133686, A355739, A355740, A367771, A367867, A367905, A370583, A370585, A370586, A370636.
%K A370593 nonn
%O A370593 0,4
%A A370593 _Gus Wiseman_, Feb 29 2024