A328171 -id:A328171 - cp's OEIS frontend

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

Gus Wiseman, Oct 29 2019

nonn

			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

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.

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

A328678 Number of strict, pairwise indivisible, relatively prime integer partitions of n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 1, 2, 2, 4, 3, 5, 4, 5, 7, 10, 9, 12, 11, 14, 15, 22, 20, 25, 26, 32, 33, 44, 41, 54, 49, 62, 67, 80, 80, 100, 100, 118, 121, 152, 148, 179, 178, 210, 219, 267, 259, 316, 313, 363, 380, 449, 448, 529, 532, 619, 640, 745, 749, 867, 889

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

Note that pairwise indivisibility implies strictness, but we include "strict" in the name in order to more clearly distinguish it from A328676 = "Number of relatively prime integer partitions of n whose distinct parts are pairwise indivisible".

			The a(1) = 1 through a(20) = 11 partitions (A..H = 10..20) (empty columns not shown):
  1  32  43  53  54  73   65  75   76  95   87   97   98    B7   A9    B9
         52      72  532  74  543  85  B3   B4   B5   A7    D5   B8    D7
                          83  732  94  743  D2   D3   B6    765  C7    H3
                          92       A3  752  654  754  C5    873  D6    875
                                   B2       753  853  D4    954  E5    965
                                                 952  E3    972  F4    974
                                                 B32  F2    B43  G3    A73
                                                      764   B52  H2    B54
                                                      A43   D32  865   B72
                                                      7532       964   D43
                                                                 B53   D52
                                                                 7543

The Heinz numbers of these partitions are the squarefree terms of A328677.
The non-strict case is A328676.
Pairwise indivisible partitions are A303362.
Strict, relatively prime partitions are A078374.
A ranking function using binary indices is A328671.
Cf. A000837, A285572, A285573, A304713, A305148, A316476, A328171.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#==1&&stableQ[#,Divisible]&]],{n,30}]

Moebius transform of A303362.

cp's OEIS Frontend

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

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