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%I #6 Nov 05 2020 22:56:20
%S 0,0,0,0,0,0,0,0,0,1,1,2,2,4,4,7,6,10,8,14,12,18,16,24,18,30,25,34,30,
%T 44,31,52,42,56,49,69,50,80,64,83,70,102,71,114,90,112,100,140,98,153,
%U 117,153,132,184,128,195,154,196,169,234,156,252,196,241
%N Number of relatively prime 3-part strict integer partitions of n with no 1's.
%C The Heinz numbers of these partitions are the intersection of A005117 (strict), A005408 (no 1's), A014612 (length 3), and A289509 (relatively prime).
%e The a(9) = 1 through a(19) = 14 triples (A = 10, B = 11, C = 12, D = 13, E = 14):
%e 432 532 542 543 643 653 654 754 764 765 865
%e 632 732 652 743 753 763 854 873 874
%e 742 752 762 853 863 954 964
%e 832 932 843 943 872 972 973
%e 852 952 953 A53 982
%e 942 B32 962 B43 A54
%e A32 A43 B52 A63
%e A52 D32 A72
%e B42 B53
%e C32 B62
%e C43
%e C52
%e D42
%e E32
%t Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,30}]
%Y A001399(n-9) does not require relative primality.
%Y A005117 /\ A005408 /\ A014612 /\ A289509 gives the Heinz numbers.
%Y A055684 is the 2-part version.
%Y A284825 counts the case that is also pairwise non-coprime.
%Y A337452 counts these partitions of any length.
%Y A337563 is the pairwise coprime instead of relatively prime version.
%Y A337605 is the pairwise non-coprime instead of relative prime version.
%Y A338332 is the not necessarily strict version.
%Y A338333*6 is the ordered version.
%Y A000837 counts relatively prime partitions.
%Y A008284 counts partitions by sum and length.
%Y A078374 counts relatively prime strict partitions.
%Y A101271 counts 3-part relatively prime strict partitions.
%Y A220377 counts 3-part pairwise coprime strict partitions.
%Y A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
%Y Cf. A000010, A000217, A000741, A023022, A082024, A302698, A307719, A337450, A337599, A338468.
%K nonn
%O 0,12
%A _Gus Wiseman_, Oct 30 2020
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