A343377 Number of strict integer partitions of n with no part divisible by all the others.

1, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 8, 9, 13, 18, 21, 26, 32, 38, 47, 57, 66, 80, 95, 110, 132, 157, 181, 211, 246, 282, 327, 379, 435, 500, 570, 648, 743, 849, 963, 1094, 1241, 1404, 1592, 1799, 2025, 2282, 2568, 2882, 3239, 3634, 4066, 4554, 5094, 5686, 6346

Offset: 0

Author

Gus Wiseman, Apr 16 2021

Keywords

• nonn

Comments

Alternative name: Number of strict integer partitions of n that are empty or have greatest part not divisible by all the others.

Examples

			The a(5) = 1 through a(12) = 9 partitions:
  (3,2)  (3,2,1)  (4,3)  (5,3)    (5,4)    (6,4)      (6,5)      (7,5)
                  (5,2)  (4,3,1)  (7,2)    (7,3)      (7,4)      (5,4,3)
                         (5,2,1)  (4,3,2)  (5,3,2)    (8,3)      (6,4,2)
                                  (5,3,1)  (5,4,1)    (9,2)      (6,5,1)
                                           (7,2,1)    (5,4,2)    (7,3,2)
                                           (4,3,2,1)  (6,4,1)    (7,4,1)
                                                      (7,3,1)    (8,3,1)
                                                      (5,3,2,1)  (9,2,1)
                                                                 (5,4,2,1)
		

Crossrefs

The dual strict complement is A097986.

The dual version is A341450.

The non-strict version is A343341 (Heinz numbers: A343337).

The strict complement is counted by A343347.

The case with smallest part not divisible by all the others is A343379.

The case with smallest part divisible by all the others is A343381.

A000005 counts divisors.

A000009 counts strict partitions.

A000070 counts partitions with a selected part.

A006128 counts partitions with a selected position.

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.

A339564 counts factorizations with a selected factor.

Cf. A083710, A130689, A200745, A264401, A338470, A339562, A343338, A343342, A343345, A343346, A343382.

Programs

Mathematica

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