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