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Number of integer partitions of n that are either constant or relatively prime.
3

%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