A343345 Number of integer partitions of n that are empty, or have smallest part dividing all the others, but do not have greatest part divisible by all the others.

1, 0, 0, 0, 0, 0, 1, 1, 4, 6, 11, 16, 29, 36, 59, 79, 115, 149, 216, 270, 379, 473, 634, 793, 1063, 1292, 1689, 2079, 2667, 3241, 4142, 4982, 6291, 7582, 9434, 11321, 14049, 16709, 20545, 24490, 29860, 35380, 43004, 50741, 61282, 72284, 86680, 101906, 121990

Offset

0,9

Comments

First differs from A343346 at a(14) = 79, A343346(14) = 80.

Alternative name: Number of integer partitions of n with a part dividing all the others, but with no part divisible by all the others.

Links

Table of n, a(n) for n=0..48.

Example

The a(6) = 1 through a(11) = 16 partitions:
  (321)  (3211)  (431)    (531)     (541)      (641)
                 (521)    (3321)    (721)      (731)
                 (3221)   (4311)    (4321)     (4331)
                 (32111)  (5211)    (5221)     (5321)
                          (32211)   (5311)     (5411)
                          (321111)  (32221)    (7211)
                                    (33211)    (33221)
                                    (43111)    (43211)
                                    (52111)    (52211)
                                    (322111)   (53111)
                                    (3211111)  (322211)
                                               (332111)
                                               (431111)
                                               (521111)
                                               (3221111)
                                               (32111111)

Mathematica

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

Crossrefs

The first condition alone gives A083710.

The half-opposite versions are A130714 and A343342.

The Heinz numbers of these partitions are 1 and A343340.

The second condition alone gives A343341.

The opposite version is A343344.

The strict case is A343381.

A000009 counts strict partitions.

A000041 counts 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).

Cf. A083711, A097986, A098743, A098965, A130689, A264401, A339563, A343337.

Sequence in context: A190564 A008369 A343341 * A343346 A296468 A060577

Adjacent sequences: A343342 A343343 A343344 * A343346 A343347 A343348

Keyword

nonn

Author

Gus Wiseman, Apr 15 2021

Status

approved