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

0,8

Comments

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

Links

Table of n, a(n) for n=0..55.

Example

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)

Mathematica

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

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.

Sequence in context: A018338 A018271 A338349 * A073667 A326497 A325046

Adjacent sequences: A343374 A343375 A343376 * A343378 A343379 A343380

Keyword

nonn

Author

Gus Wiseman, Apr 16 2021

Status

approved