A337485 Number of pairwise coprime integer partitions of n with no 1's, where a singleton is not considered coprime unless it is (1).

0, 0, 0, 0, 0, 1, 0, 2, 1, 2, 2, 4, 3, 5, 4, 4, 7, 8, 9, 10, 10, 9, 13, 17, 18, 17, 19, 19, 24, 29, 34, 33, 31, 31, 42, 42, 56, 55, 50, 54, 66, 77, 86, 86, 79, 81, 96, 124, 127, 126, 127, 126, 145, 181, 190, 184, 183, 192, 212, 262, 289, 278, 257, 270, 311

Offset: 0

Author

Gus Wiseman, Sep 21 2020

Keywords

• nonn

Comments

Such a partition is necessarily strict.

The Heinz numbers of these partitions are the intersection of A005408 (no 1's), A005117 (strict), and A302696 (coprime).

Examples

			The a(n) partitions for n = 5, 7, 12, 13, 16, 17, 18, 19 (A..H = 10..17):
  (3,2)  (4,3)  (7,5)    (7,6)  (9,7)    (9,8)      (B,7)    (A,9)
         (5,2)  (5,4,3)  (8,5)  (B,5)    (A,7)      (D,5)    (B,8)
                (7,3,2)  (9,4)  (D,3)    (B,6)      (7,6,5)  (C,7)
                         (A,3)  (7,5,4)  (C,5)      (8,7,3)  (D,6)
                         (B,2)  (8,5,3)  (D,4)      (9,5,4)  (E,5)
                                (9,5,2)  (E,3)      (9,7,2)  (F,4)
                                (B,3,2)  (F,2)      (B,4,3)  (G,3)
                                         (7,5,3,2)  (B,5,2)  (H,2)
                                                    (D,3,2)  (B,5,3)
                                                             (7,5,4,3)
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750

Crossrefs

A005408 intersected with A302696 ranks these partitions.

A007359 considers all singletons to be coprime.

A327516 allows 1's, with non-strict version A305713.

A337452 is the relatively prime instead of pairwise coprime version, with non-strict version A302698.

A337563 is the restriction to partitions of length 3.

A002865 counts partitions with no 1's.

A078374 counts relatively prime strict partitions.

A200976 and A328673 count pairwise non-coprime partitions.

Cf. A101268, A220377, A302696, A304709, A332004, A337450, A337451, A337462, A337561, A337605.

Programs

Mathematica

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

Formula

a(n) = A007359(n) - 1 for n > 1.