A343378 Number of strict integer partitions of n that are empty or such that (1) the smallest part divides every other part and (2) the greatest part is divisible by every other part.

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 3, 6, 5, 4, 6, 6, 4, 8, 6, 7, 9, 8, 5, 12, 9, 8, 9, 11, 6, 14, 10, 10, 11, 10, 10, 20, 12, 12, 15, 18, 10, 21, 13, 15, 19, 17, 11, 27, 19, 20, 20, 25, 13, 27, 22, 26, 23, 24, 15, 34, 23, 21, 27, 30, 19, 38, 24, 26, 27, 37

Offset: 0

Author

Gus Wiseman, Apr 16 2021

Keywords

• nonn

Comments

Alternative name: Number of strict integer partitions of n with a part dividing all the others and a part divisible by all the others.

Examples

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3   4   5   6   7    8   9    A    B    C     D    E    F
        21  31  41  42  61   62  63   82   A1   84    C1   C2   A5
                    51  421  71  81   91   821  93    841  D1   C3
                                 621  631       A2    931  842  E1
                                                B1    A21       C21
                                                6321            8421
		

Crossrefs

The first condition alone gives A097986.

The non-strict version is A130714 (Heinz numbers are complement of A343343).

The second condition alone gives A343347.

The opposite version is A343379.

The half-opposite versions are A343380 and A343381.

The strict complement is counted by A343382.

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, A264401, A339562, A339563, A341450, A343346, A343377.

Programs

Mathematica

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