A328171 Number of (necessarily strict) integer partitions of n with no two consecutive parts divisible.

1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 4, 5, 4, 9, 9, 10, 12, 14, 16, 20, 23, 29, 34, 38, 41, 51, 60, 66, 78, 89, 103, 119, 137, 157, 180, 201, 229, 261, 298, 338, 379, 431, 486, 547, 618, 694, 783, 876, 986, 1103, 1241, 1387, 1551, 1728, 1932, 2148, 2395, 2664, 2963

Offset

0,6

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..330

Example

The a(1) = 1 through a(15) = 10 partitions (A..F = 10..15):
  1  2  3  4  5   6  7   8   9    A    B   C    D    E     F
              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
                                                B2   653   654
                                                643  743   753
                                                652  752   852
                                                832  5432  A32
                                                           6432

Mathematica

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

Crossrefs

The complement is counted by A328221.

The Heinz numbers of these partitions are A328603.

Partitions whose pairs of consecutive parts are relatively prime are A328172, with strict case A328188.

Partitions with no pair of consecutive parts relatively prime are A328187, with strict case A328220.

Numbers without consecutive divisible proper divisors are A328028.

Cf. A000837, A018783, A328026, A328161, A328189, A328194, A328195.

Sequence in context: A183912 A144693 A328399 * A029139 A100927 A001687

Adjacent sequences: A328168 A328169 A328170 * A328172 A328173 A328174

Keyword

nonn

Author

Gus Wiseman, Oct 11 2019

Status

approved