A328171 - OEIS

%I #13 Jan 10 2021 08:54:44

%S 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,

%T 78,89,103,119,137,157,180,201,229,261,298,338,379,431,486,547,618,

%U 694,783,876,986,1103,1241,1387,1551,1728,1932,2148,2395,2664,2963

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

%H Fausto A. C. Cariboni, <a href="/A328171/b328171.txt">Table of n, a(n) for n = 0..330</a>

%e The a(1) = 1 through a(15) = 10 partitions (A..F = 10..15):

%e   1  2  3  4  5   6  7   8   9    A    B   C    D    E     F

%e               32     43  53  54   64   65  75   76   86    87

%e                      52      72   73   74  543  85   95    96

%e                              432  532  83  732  94   A4    B4

%e                                        92       A3   B3    D2

%e                                                 B2   653   654

%e                                                 643  743   753

%e                                                 652  752   852

%e                                                 832  5432  A32

%e                                                            6432

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

%Y The complement is counted by A328221.

%Y The Heinz numbers of these partitions are A328603.

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

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

%Y Numbers without consecutive divisible proper divisors are A328028.

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

%K nonn

%O 0,6

%A _Gus Wiseman_, Oct 11 2019