A338332 Number of relatively prime 3-part integer partitions of n with no 1's.

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 3, 8, 6, 9, 9, 16, 10, 21, 15, 22, 20, 33, 21, 38, 30, 41, 35, 56, 34, 65, 49, 64, 56, 79, 55, 96, 72, 93, 77, 120, 76, 133, 99, 122, 110, 161, 105, 172, 126, 167, 143, 208, 136, 213, 165, 212, 182, 261, 163, 280, 210, 257

Offset: 0

Author

Gus Wiseman, Oct 30 2020

Keywords

• nonn

Comments

The Heinz numbers of these partitions are the intersection of A005408 (no 1's), A014612 (length 3), and A289509 (relatively prime).

Examples

			The a(7) = 1 through a(17) = 16 triples (A = 10, B = 11, C = 12, D = 13):
  322   332   432   433   443   543   544   554   654   655   665
              522   532   533   552   553   653   744   754   755
                          542   732   643   743   753   763   764
                          632         652   752   762   772   773
                          722         733   833   843   853   854
                                      742   932   852   943   863
                                      832         942   952   872
                                      922         A32   A33   944
                                                  B22   B32   953
                                                              962
                                                              A43
                                                              A52
                                                              B33
                                                              B42
                                                              C32
                                                              D22
		

Crossrefs

A001399(n-6) does not require relative primality.

A005408 /\ A014612 /\ A289509 gives the Heinz numbers of these partitions.

A055684 is the 2-part version.

A284825 counts the case that is also pairwise non-coprime.

A302698 counts these partitions of any length.

A337563 is the pairwise coprime instead of relatively prime version.

A338333 is the strict version.

A000837 counts relatively prime partitions, with strict case A078374.

A008284 counts partitions by sum and length.

Cf. A000010, A000741, A023022, A078374, A082024, A101271, A307719, A337450, A337599, A337600, A337601.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n,{3}],!MemberQ[#,1]&&GCD@@#==1&]],{n,0,30}]