A343381 Number of strict integer partitions of n with a part dividing all the others but no part divisible by all the others.

1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 4, 9, 9, 14, 14, 20, 20, 30, 30, 39, 44, 59, 59, 77, 85, 106, 114, 145, 150, 191, 205, 247, 267, 328, 345, 418, 455, 544, 582, 699, 745, 886, 962, 1117, 1209, 1430, 1523, 1778, 1932, 2225, 2406, 2792, 3001, 3456, 3750

Offset: 0

Author

Gus Wiseman, Apr 16 2021

Keywords

• nonn

Comments

Alternative name: Number of strict integer partitions of n that are empty or (1) have smallest part dividing all the others and (2) have greatest part not divisible by all the others.

Examples

			The a(6) = 1 through a(16) = 14 partitions (empty column indicated by dot, A..D = 10..13):
  321   .  431   531   541    641    642    751    761    861     862
           521         721    731    651    5431   851    951     871
                       4321   5321   741    6421   941    A41     961
                                     831    7321   A31    B31     A42
                                     921           B21    6531    B41
                                     5421          6431   7431    D21
                                                   6521   7521    6541
                                                   7421   9321    7531
                                                   8321   54321   7621
                                                                  8431
                                                                  8521
                                                                  9421
                                                                  A321
                                                                  64321
		

Crossrefs

The first condition alone gives A097986.

The non-strict version is A343345 (Heinz numbers: A343340).

The second condition alone gives A343377.

The half-opposite versions are A343378 and A343379.

The opposite (and dual) version is A343380.

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, A339563, A341450, A343337, A343341, A343347, A343382.

Programs

Mathematica

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