A341450 Number of strict integer partitions of n that are empty or have smallest part not dividing all the others.

1, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 3, 9, 9, 12, 12, 20, 18, 28, 27, 37, 42, 55, 51, 74, 80, 98, 105, 136, 137, 180, 189, 232, 255, 308, 320, 403, 434, 512, 551, 668, 706, 852, 915, 1067, 1170, 1370, 1453, 1722, 1860, 2145, 2332, 2701, 2899, 3355, 3626, 4144

Offset: 0

Author

Gus Wiseman, Apr 15 2021

Keywords

• nonn

Comments

Alternative name: Number of strict integer partitions of n with no part dividing all the others.

Examples

			The a(0) = 1 through a(15) = 12 strict partitions (empty columns indicated by dots, 0 represents the empty partition, A..D = 10..13):
  0  .  .  .  .  32   .  43   53   54    64    65    75    76    86     87
                         52        72    73    74    543   85    95     96
                                   432   532   83    732   94    A4     B4
                                               92          A3    B3     D2
                                               542         B2    653    654
                                               632         643   743    753
                                                           652   752    762
                                                           742   932    843
                                                           832   5432   852
                                                                        942
                                                                        A32
                                                                        6432
		

Crossrefs

The complement is counted by A097986 (non-strict: A083710, rank: A339563).

The complement with no 1's is A098965 (non-strict: A083711).

The non-strict version is A338470.

The Heinz numbers of these partitions are A339562 (non-strict: A342193).

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

The case with greatest part divisible by all others is A343380.

A000009 counts strict partitions (non-strict: A000041).

A000070 counts partitions with a selected part.

A006128 counts partitions with a selected position.

A015723 counts strict partitions with a selected part.

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

Sequences with similar formulas: A024994, A047966, A047968, A168111.

Cf. A001787, A001792, A064410, A264401, A343342, A343381, A343382.

Programs

Mathematica

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

Formula

a(n > 0) = A000009(n) - Sum_{d|n} A025147(d-1).