A328675 Number of integer partitions of n with no two distinct consecutive parts divisible.

1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 13, 13, 22, 23, 30, 36, 50, 54, 77, 85, 113, 135, 170, 194, 256, 303, 369, 440, 545, 640, 792, 931, 1132, 1347, 1616, 1909, 2295, 2712, 3225, 3799, 4519, 5310, 6278, 7365, 8675, 10170, 11928, 13940, 16314, 19046, 22223, 25856

Offset: 0

Author

Gus Wiseman, Oct 29 2019

Keywords

• nonn

Examples

			The a(1) = 1 through a(10) = 9 partitions (A = 10).
  1  2   3    4     5      6       7        8         9          A
     11  111  22    32     33      43       44        54         55
              1111  11111  222     52       53        72         64
                           111111  322      332       333        73
                                   1111111  2222      432        433
                                            11111111  522        532
                                                      3222       3322
                                                      111111111  22222
                                                                 1111111111
		

Crossrefs

The Heinz numbers of these partitions are given by A328674.

The case involving all consecutive parts (not just distinct) is A328171.

The version for relative primality instead of divisibility is A328187.

Partitions with all consecutive parts divisible are A003238.

Compositions without consecutive divisibilities are A328460.

Cf. A305148, A316476, A318726, A328172, A328508, A328593, A328598, A328603.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!MatchQ[Union[#],{_,x_,y_,_}/;Divisible[y,x]]&]],{n,0,30}]