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

0,8

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).

Links

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

Formula

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

Example

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)

Mathematica

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

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.

Sequence in context: A288310 A332718 A341461 * A328678 A184199 A262346

Adjacent sequences: A337482 A337483 A337484 * A337486 A337487 A337488

Keyword

nonn

Author

Gus Wiseman, Sep 21 2020

Status

approved