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Number of non-constant integer partitions of n whose parts have a common divisor > 1.
2

%I #10 Nov 09 2020 21:35:22

%S 0,0,0,0,0,0,1,0,2,1,5,0,9,0,13,6,18,0,33,0,40,14,54,0,87,5,99,27,133,

%T 0,211,0,226,55,295,18,443,0,488,100,637,0,912,0,1000,198,1253,0,1775,

%U 13,1988,296,2434,0,3358,59,3728,489,4563,0,6241,0,6840,814

%N Number of non-constant integer partitions of n whose parts have a common divisor > 1.

%F For n > 0, a(n) = A018783(n) - A000005(n) + 1.

%e The a(6) = 2 through a(15) = 6 partitions (empty columns indicated by dots, A = 10, B = 11, C = 12):

%e (42) . (62) (63) (64) . (84) . (86) (96)

%e (422) (82) (93) (A4) (A5)

%e (442) (A2) (C2) (C3)

%e (622) (633) (644) (663)

%e (4222) (642) (662) (933)

%e (822) (842) (6333)

%e (4422) (A22)

%e (6222) (4442)

%e (42222) (6422)

%e (8222)

%e (44222)

%e (62222)

%e (422222)

%t Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&GCD@@#>1&]],{n,0,30}]

%Y A046022 lists positions of zeros.

%Y A082023(n) - A059841(n) is the 2-part version, n > 2.

%Y A303280(n) - 1 is the strict case (n > 1).

%Y A338552 lists the Heinz numbers of these partitions.

%Y A338553 counts the complement, with Heinz numbers A338555.

%Y A000005 counts constant partitions, with Heinz numbers A000961.

%Y A000837 counts relatively prime partitions, with Heinz numbers A289509.

%Y A018783 counts non-relatively prime partitions (ordered: A178472), with Heinz numbers A318978.

%Y A282750 counts relatively prime partitions by sum and length.

%Y Cf. A000010, A008284, A051424, A082024, A289508, A302698, A304712.

%K nonn

%O 0,9

%A _Gus Wiseman_, Nov 07 2020