%I #11 Nov 05 2020 22:57:03
%S 1,1,2,3,5,7,10,15,20,29,37,56,68,101,122,170,213,297,352,490,587,778,
%T 948,1255,1488,1953,2337,2983,3585,4565,5393,6842,8123,10088,12015,
%U 14865,17534,21637,25527,31085,36701,44583,52262,63261,74175,88936,104305,124754
%N Number of integer partitions of n that are either constant or relatively prime.
%C The Heinz numbers of these partitions are given by A338555 = A000961 \/ A289509. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
%F For n > 0, a(n) = A000005(n) + A000837(n) - 1.
%e The a(1) = 1 through a(7) = 15 partitions:
%e (1) (2) (3) (4) (5) (6) (7)
%e (11) (21) (22) (32) (33) (43)
%e (111) (31) (41) (51) (52)
%e (211) (221) (222) (61)
%e (1111) (311) (321) (322)
%e (2111) (411) (331)
%e (11111) (2211) (421)
%e (3111) (511)
%e (21111) (2221)
%e (111111) (3211)
%e (4111)
%e (22111)
%e (31111)
%e (211111)
%e (1111111)
%t Table[Length[Select[IntegerPartitions[n],SameQ@@#||GCD@@#==1&]],{n,0,30}]
%Y A023022(n) + A059841(n) is the 2-part version.
%Y A078374(n) + 1 is the strict case (n > 1).
%Y A338554 counts the complement, with Heinz numbers A338552.
%Y A338555 gives the Heinz numbers of these partitions.
%Y A000005 counts constant partitions, with Heinz numbers A000961.
%Y A000837 counts relatively prime partitions, with Heinz numbers A289509.
%Y A282750 counts relatively prime partitions by sum and length.
%Y Cf. A000010, A007360, A008284, A023023, A051424, A101271, A101391, A302698, A304712, A327516, A337664.
%K nonn
%O 0,3
%A _Gus Wiseman_, Nov 03 2020