A326843 Number of integer partitions of n whose length and maximum both divide n.

1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 22, 2, 5, 11, 16, 2, 36, 2, 46, 22, 5, 2, 209, 3, 5, 42, 130, 2, 434, 2, 217, 77, 5, 52, 1400, 2, 5, 135, 1749, 2, 1782, 2, 957, 2151, 5, 2, 8355, 3, 1859, 385, 2388, 2, 6726, 2765, 10641, 627, 5, 2, 68049, 2, 5, 13424, 17142

Offset

0,3

Comments

The Heinz numbers of these partitions are given by A326837.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..180

Example

The a(1) = 1 through a(8) = 5 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (1111)           (222)                (2222)
                                     (321)                (4211)
                                     (111111)             (11111111)
The a(12) = 22 partitions:
  (12)
  (6,6)
  (4,4,4)
  (6,3,3)
  (6,4,2)
  (6,5,1)
  (3,3,3,3)
  (4,3,3,2)
  (4,4,2,2)
  (4,4,3,1)
  (6,2,2,2)
  (6,3,2,1)
  (6,4,1,1)
  (2,2,2,2,2,2)
  (3,2,2,2,2,1)
  (3,3,2,2,1,1)
  (3,3,3,1,1,1)
  (4,2,2,2,1,1)
  (4,3,2,1,1,1)
  (4,4,1,1,1,1)
  (6,2,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)

Mathematica

Table[If[n==0, 1, Length[Select[IntegerPartitions[n], Divisible[n, Length[#]]&&Divisible[n, Max[#]]&]]], {n, 0, 30}]

Crossrefs

The strict case is A326851.

The non-constant case is A326852.

The case where all parts (not just the maximum) divide n is A326842.

Cf. A018818, A047993, A067538, A326837, A326849.

Sequence in context: A342086 A272209 A326842 * A323347 A322900 A238791

Adjacent sequences: A326840 A326841 A326842 * A326844 A326845 A326846

Keyword

nonn

Author

Gus Wiseman, Jul 26 2019

Status

approved