A339660 Number of strict integer partitions of n with no 1's and a part divisible by all the other parts.

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 1, 3, 3, 3, 3, 5, 2, 5, 5, 4, 5, 7, 3, 5, 6, 5, 5, 9, 4, 7, 6, 6, 9, 11, 6, 9, 10, 9, 10, 12, 6, 11, 12, 10, 12, 16, 9, 15, 16, 12, 14, 18, 14, 16, 18, 14, 15, 22, 11, 16, 20, 13, 21, 23, 15, 21, 24, 21, 21, 31, 14, 24

Offset: 0

Author

Gus Wiseman, Apr 19 2021

Keywords

• nonn

Comments

Alternative name: Number of strict integer partitions of n with no 1's that are empty or have greatest part divisible by all the other parts.

Examples

			The a(n) partitions for n = 14, 12, 18, 24, 30, 39, 36:
  (14)     (12)    (18)      (24)        (30)        (39)          (36)
  (12,2)   (8,4)   (12,6)    (16,8)      (24,6)      (36,3)        (27,9)
  (8,4,2)  (9,3)   (15,3)    (18,6)      (25,5)      (26,13)       (30,6)
           (10,2)  (16,2)    (20,4)      (27,3)      (27,9,3)      (32,4)
                   (12,4,2)  (21,3)      (28,2)      (28,7,4)      (33,3)
                             (22,2)      (20,10)     (30,6,3)      (34,2)
                             (12,6,4,2)  (18,9,3)    (24,12,3)     (24,12)
                                         (24,4,2)    (24,8,4,3)    (24,8,4)
                                         (16,8,4,2)  (20,10,5,4)   (18,9,6,3)
                                                     (24,6,4,3,2)  (24,6,4,2)
                                                                   (20,10,4,2)
		

Crossrefs

The dual version is A098965 (non-strict: A083711).

The non-strict version is A339619 (Heinz numbers: complement of A343337).

The version with 1's allowed is A343347 (non-strict: A130689).

The case without a part dividing all the other parts is A343380.

A000009 counts strict partitions.

A000070 counts partitions with a selected part.

A015723 counts strict partitions with a selected part.

A018818 counts partitions into divisors (strict: A033630).

A167865 counts strict chains of divisors > 1 summing to n.

Cf. A083710, A097986, A098743, A200745, A264401, A339563, A341450, A343337, A343341, A343344, A343378.

Programs

Mathematica

Table[If[n==0,1,Length[Select[IntegerPartitions[n],FreeQ[#,1]&&UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]]],{n,0,30}]