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

Author

Gus Wiseman, Jul 26 2019

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A326837.

Examples

			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)
		

Links

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

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.

Programs

Mathematica

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