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%I #10 Oct 28 2023 23:53:01
%S 0,0,0,1,0,2,1,4,1,8,3,13,6,21,10,36,15,53,28,80,41,122,63,174,97,250,
%T 140,359,201,496,299,685,410,949,575,1284,804,1726,1093,2327,1482,
%U 3076,2023,4060,2684,5358,3572,6970,4745,9050,6221,11734,8115,15060,10609
%N Number of integer partitions of n whose odd parts have a common divisor > 1.
%e The a(3) = 1 through a(11) = 13 partitions:
%e (3) . (5) (3,3) (7) (3,3,2) (9) (5,5) (11)
%e (3,2) (4,3) (5,4) (4,3,3) (6,5)
%e (5,2) (6,3) (3,3,2,2) (7,4)
%e (3,2,2) (7,2) (8,3)
%e (3,3,3) (9,2)
%e (4,3,2) (4,4,3)
%e (5,2,2) (5,4,2)
%e (3,2,2,2) (6,3,2)
%e (7,2,2)
%e (3,3,3,2)
%e (4,3,2,2)
%e (5,2,2,2)
%e (3,2,2,2,2)
%t Table[Length[Select[IntegerPartitions[n], GCD@@Select[#,OddQ]>1&]], {n,0,30}]
%o (Python)
%o from math import gcd
%o from sympy.utilities.iterables import partitions
%o def A366842(n): return sum(1 for p in partitions(n) if gcd(*(q for q in p if q&1))>1) # _Chai Wah Wu_, Oct 28 2023
%Y This is the odd case of A018783, complement A000837.
%Y The even version is A047967.
%Y The complement is counted by A366850, ranks A366846.
%Y A000041 counts integer partitions, strict A000009.
%Y A000740 counts relatively prime compositions.
%Y A113685 counts partitions by sum of odds, stat A366528, w/o zeros A365067.
%Y A168532 counts partitions by gcd.
%Y A239261 counts partitions with (sum of odd parts) = (sum of even parts).
%Y A289508 gives gcd of prime indices, positions of ones A289509.
%Y Cf. A007359, A051424, A055922, A066208, A078374, A087436, A116598, A337485, A366843, A366844, A366845.
%K nonn
%O 0,6
%A _Gus Wiseman_, Oct 28 2023
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