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%I #6 Oct 30 2023 11:06:31
%S 0,1,1,2,3,5,7,11,16,22,32,43,60,80,110,140,194,244,327,410,544,670,
%T 883,1081,1401,1708,2195,2651,3382,4069,5129,6157,7708,9194,11438,
%U 13599,16788,19911,24432,28858,35229,41507,50359,59201,71489,83776,100731,117784
%N Number of integer partitions of n whose odd parts are relatively prime.
%e The a(1) = 1 through a(8) = 16 partitions:
%e (1) (11) (21) (31) (41) (51) (61) (53)
%e (111) (211) (221) (321) (331) (71)
%e (1111) (311) (411) (421) (431)
%e (2111) (2211) (511) (521)
%e (11111) (3111) (2221) (611)
%e (21111) (3211) (3221)
%e (111111) (4111) (3311)
%e (22111) (4211)
%e (31111) (5111)
%e (211111) (22211)
%e (1111111) (32111)
%e (41111)
%e (221111)
%e (311111)
%e (2111111)
%e (11111111)
%t Table[Length[Select[IntegerPartitions[n],GCD@@Select[#,OddQ]==1&]],{n,0,30}]
%Y For all parts (not just odd) we have A000837, complement A018783.
%Y The complement is counted by A366842.
%Y These partitions have ranks A366846.
%Y A000041 counts integer partitions, strict A000009 (also into odds).
%Y A000740 counts relatively prime compositions.
%Y A078374 counts relatively prime strict partitions.
%Y A113685 counts partitions by sum of odd parts, rank statistic A366528.
%Y A168532 counts partitions by gcd.
%Y A239261 counts partitions with (sum of odd parts) = (sum of even parts).
%Y Cf. A007359, A047967, A051424, A066208, A116598, A365067, A366843, A366844, A366845, A366848.
%K nonn
%O 0,4
%A _Gus Wiseman_, Oct 28 2023
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